Regressão Linear

### Modelos estatísticos - Aula de modelos lineares
### Pacotes usados na aula.
require("ISLR")
## Carregando pacotes exigidos: ISLR
require("ggplot2")
## Carregando pacotes exigidos: ggplot2
require("GGally")
## Carregando pacotes exigidos: GGally
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
require("leaps")
## Carregando pacotes exigidos: leaps
require("car")
## Carregando pacotes exigidos: car
## Carregando pacotes exigidos: carData
options(device = X11)

Recomendações de leitura

[…]

Objetivo

Num modelo de regressão linear, a relação entre uma variável aleatória de interesse (variável resposta) e um conjunto de preditores (variáveis explicativas) é definida por uma função linear envolvendo os preditores e um conjunto de parâmetros associados (coeficientes ou $$′s ou parametros).

“Composição”

  • Variaveis Resposta
  • Variaveis Explicativas

Exemplo:

  • Aplicação 1: Preço de Imóveis
    • Variável Resposta: Preço do imóvel
    • Variáveis Explicativas: Tamanho do imóvel (m2), número de quartos, localização (bairro), idade do imóvel, presença de garagem
  • Aplicação 2: Desempenho Escolar
    • Variável Resposta: Nota final em uma disciplina
    • Variáveis Explicativas: Número de horas estudadas por semana, frequência nas aulas, nota em provas anteriores, participação em atividades extraclasse, apoio familar
  • Aplicação 3: Vendas de um produto
    • Variável Resposta: Vendas mensais de um produto
    • Variáveis Explicativas: Preço do produto, gastos em marketing, número de vendedores, promoções realizadas, condições econômicas.
  • Aplicação 4: Consumo de Energia
    • Variável Resposta: Consumo de energia elétrica (kWh)
    • Variáveis Explicativas: Tamanho da residência (m2), número de moradores, uso de eletrodomésticos, temperatura média na região, tipo de aquecimento
  • Aplicação 5: Rendimento de Investimentos
    • Variável Resposta: Rendimento de uma carteira de investimentos
    • Variáveis Explicativas: Taxa de juros, porcentagem de ações vs. renda fixa, valor inicial investido, taxa de inflação, duração do investimento

Modelo de regressão linear

Seja \(y\) a variável resposta e \(x1, x2, ..., xk\) os \(k\) preditores. O modelo de regressão linear fica definido por: \[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x_k + \epsilon \]

onde:

  • \(\beta_0\) é o intercepto do modelo (se todos os \(x\)s e \(\epsilon\) forem 0 $ y = _0$ );
  • \(\beta_1, \beta_2, ... \beta_k\) são parâmetroa do modelo, que correspondem aos efeitos lineares de cada um dos \(k\) preditores;
  • \(\epsilon\) representa os erros aleatórios, específicos para cada indivíduo. É a parte de \(y\) que não é explicada pela regressão linear.

Erros

Embora sejam não observáveis e não explicados pelo modelo, na regressão linear assumimos as seguintes propriedades para os erros:

  • Os erros têm média zero;
  • Os erros têm variância constante;
  • Os erros para dois indivíduos quaisquer são não correlacionados;
  • Os erros têm distribuição normal.

Este conjunto de suposições é usualmente denotado por \(\epsilon \sim N (0, \sigma^2)\).

Regressão Linear Simples

Como caso particular da regressão linear temos a regressão linear simples, que se caracteriza por considerar um único preditor:

\[ y = \beta_0 + \beta_1 x + \epsilon, \space \space \space \space \space \epsilon \sim N (0, \sigma^2) \]

Regressão Linear Polinomial

Outro modelo de regressão linear bastante usual é o modelo de regressão polinomial, que permite explicar uma relação não linear entre a resposta e o(s) preditor(es):

\[ y = \beta_0 + \beta_1 x + \beta_2 x^2 + ... + \beta_k x^k + \epsilon, \space \space \space \space \space \epsilon \sim N (0, \sigma^2) \] Fig 01

Fig 02
Fig 02
Fig 03
Fig 03
Fig 04
Fig 04
Fig 05
Fig 05
Fig 06
Fig 06

Um breve histórico

  • Século 19: Desenvolvimento da teoria de mínimos quadrados, que serve de base para o ajuste de modelos de regressão.
  • A teoria de mínimos quadrados teve origem na Física motivada, dentre outros, por problemas na área navegação (século 18).
  • Ao longo do século 19, modelos lineares e o método de mínimos quadrados passaram a ser utilizados em outras ciências, baseados em modelos pré-estabelecidos, ou apenas em evidências empíricas.
  • O termo regressão foi introduzido por Francis Galton em 1875, baseado no princípio da “regressão à média”.

Regressão linear

  • Os dois objetivos principais de uma análise de regressão são os seguintes:
    • Exploratório- Analisar a relação funcional entre uma variável aleatória (resposta) e um conjunto de variáveis preditoras.
    • Preditivo- Predizer valores não observados da variável resposta para valores especificados das variáveis preditoras.

Exemplo- Consumo de combustível

  • Neste exemplo vamos analisar os dados da base Auto, disponíveis na biblioteca ISLR do R.
  • A base de dados dispõe de informações técnicas de 392 modelos de automóveis das décadas de 1970 e 1980, como consumo de combustível. potência do motor, dimensões dentre outras.
  • O objetivo aqui é ajustar modelos de regressão linear que expliquem o consumo de combustível com base em características do modelo.
  • Neste primeiro momento, para fins ilustrativos vamos considerar apenas duas variáveis de cada vez, embora o mais usual seja considerar múltiplas variáveis conjuntamente numa análise de regressão.
summary(Auto)
##       mpg          cylinders      displacement     horsepower        weight    
##  Min.   : 9.00   Min.   :3.000   Min.   : 68.0   Min.   : 46.0   Min.   :1613  
##  1st Qu.:17.00   1st Qu.:4.000   1st Qu.:105.0   1st Qu.: 75.0   1st Qu.:2225  
##  Median :22.75   Median :4.000   Median :151.0   Median : 93.5   Median :2804  
##  Mean   :23.45   Mean   :5.472   Mean   :194.4   Mean   :104.5   Mean   :2978  
##  3rd Qu.:29.00   3rd Qu.:8.000   3rd Qu.:275.8   3rd Qu.:126.0   3rd Qu.:3615  
##  Max.   :46.60   Max.   :8.000   Max.   :455.0   Max.   :230.0   Max.   :5140  
##                                                                                
##   acceleration        year           origin                      name    
##  Min.   : 8.00   Min.   :70.00   Min.   :1.000   amc matador       :  5  
##  1st Qu.:13.78   1st Qu.:73.00   1st Qu.:1.000   ford pinto        :  5  
##  Median :15.50   Median :76.00   Median :1.000   toyota corolla    :  5  
##  Mean   :15.54   Mean   :75.98   Mean   :1.577   amc gremlin       :  4  
##  3rd Qu.:17.02   3rd Qu.:79.00   3rd Qu.:2.000   amc hornet        :  4  
##  Max.   :24.80   Max.   :82.00   Max.   :3.000   chevrolet chevette:  4  
##                                                  (Other)           :365
ggplot(Auto, aes(x = year)) + geom_histogram()
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

ggplot(Auto, aes(x = year, y = mpg)) + geom_point() +
    stat_smooth(method = "lm") +
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

plot1 <- ggplot(Auto, aes(x = horsepower, y = mpg)) + geom_point() + theme_bw(base_size = 14)
plot1

plot1 + coord_trans(x="log2", y="log2")

Fig 07
Fig 07
  • Inicialmente, vamos ajustar um modelo de regressão linear que permita explicar o consumo de combustível (mpg- variável resposta) em função do ano de lançamento do modelo (yearvariável explicativa).

  • A Figura 7 sugere relação linear crescente entre o consumo de combustível e o ano de lançamento do modelo.

  • O modelo de regressão linear para esse par de variáveis fica especificado por: \[ mpg = \beta_0 + \beta_1 \times year + \epsilon, \] onde \(\beta_0\) e \(\beta_1\) são os parâmetros do modelo (intercepto e inclinação da reta de regressão) e \(\epsilon\) representa os erros aleatórios.

  • O ajuste da regressão linear consiste na estimação dos parâmetros do modelo (β0 e β1), com base nos dados amostrais, que produzem a reta de regressão que melhor se ajusta aos dados.

  • O método usual de estimação dos parâmetros de uma regressão linear é o método de mínimos quadrados, que será estudado adiante.

  • Aplicando o método de mínimos quadrados, obtemos os parâmetros estimados \(\hat{\beta_0} = −70.01\) e \(\hat{\beta_1} = 1.23\), produzindo a seguinte reta de regressão ajustada: \[ \hat{mpg} = −70.01 + 1.23 \times year \]

Fig 08
Fig 08
Fig 09
Fig 09

[continua….]

Exemplo prático- Taxa de formados

  • Na parte prática desta aula vamos analisar dados sobre a taxa de formados em \(n=777\) universidades norte-americanas no ano de 1995.
  • O objetivo é ajustar um modelo de regressão linear que permita explicar a taxa de alunos formados em função das características das universidades e de seus alunos.
  • Uma breve descrição das variáveis explicativas é apresentada na sequência.

Variaveis

  • Grad.Rate: Taxa de alunos formados (resposta);
  • Apps: Número de alunos inscritos;
  • Accept: Número de alunos aceitos;
  • Enroll: Número de novos alunos matriculados;
  • Top10perc: Percentual de novos estudantes entre os 10% melhores no ensino médio;
  • Top25perc: Percentual de novos estudantes entre os 25% melhores no ensino médio;
  • F.Undergrad: Número de alunos em período integral nos cursos de graduação;
  • P.Undergrad: Número de alunos em período parcial nos cursos de graduação;
  • Outstate: Número de alunos bolsistas de outros estados;
  • Room.Board: Gastos de hospedagem e alimentação;
  • Books: Gastos com materiais bibliográficos;
  • Personal: Gastos com recursos humanos;
  • PhD: Percentual de professores com doutorado;
  • Terminal: Percentual de professores com grau terminal;
  • S.F.Ratio: Razão alunos/professor;
  • perc.alumni: Percentual de ex-alunos que contribuem com donativos;
  • Expend: Gasto educacional por aluno.
  • Privada ou publica

CODIGO

### Carregamento e visualização inicial da base

data("College") ### Carregando a base
#help("College") ### Acessando a documentação

head(College,10) ### Visualizando as dez primeiras linhas
##                              Private Apps Accept Enroll Top10perc Top25perc
## Abilene Christian University     Yes 1660   1232    721        23        52
## Adelphi University               Yes 2186   1924    512        16        29
## Adrian College                   Yes 1428   1097    336        22        50
## Agnes Scott College              Yes  417    349    137        60        89
## Alaska Pacific University        Yes  193    146     55        16        44
## Albertson College                Yes  587    479    158        38        62
## Albertus Magnus College          Yes  353    340    103        17        45
## Albion College                   Yes 1899   1720    489        37        68
## Albright College                 Yes 1038    839    227        30        63
## Alderson-Broaddus College        Yes  582    498    172        21        44
##                              F.Undergrad P.Undergrad Outstate Room.Board Books
## Abilene Christian University        2885         537     7440       3300   450
## Adelphi University                  2683        1227    12280       6450   750
## Adrian College                      1036          99    11250       3750   400
## Agnes Scott College                  510          63    12960       5450   450
## Alaska Pacific University            249         869     7560       4120   800
## Albertson College                    678          41    13500       3335   500
## Albertus Magnus College              416         230    13290       5720   500
## Albion College                      1594          32    13868       4826   450
## Albright College                     973         306    15595       4400   300
## Alderson-Broaddus College            799          78    10468       3380   660
##                              Personal PhD Terminal S.F.Ratio perc.alumni Expend
## Abilene Christian University     2200  70       78      18.1          12   7041
## Adelphi University               1500  29       30      12.2          16  10527
## Adrian College                   1165  53       66      12.9          30   8735
## Agnes Scott College               875  92       97       7.7          37  19016
## Alaska Pacific University        1500  76       72      11.9           2  10922
## Albertson College                 675  67       73       9.4          11   9727
## Albertus Magnus College          1500  90       93      11.5          26   8861
## Albion College                    850  89      100      13.7          37  11487
## Albright College                  500  79       84      11.3          23  11644
## Alderson-Broaddus College        1800  40       41      11.5          15   8991
##                              Grad.Rate
## Abilene Christian University        60
## Adelphi University                  56
## Adrian College                      54
## Agnes Scott College                 59
## Alaska Pacific University           15
## Albertson College                   55
## Albertus Magnus College             63
## Albion College                      73
## Albright College                    80
## Alderson-Broaddus College           52
dim(College) ### Acessando a dimensão da base
## [1] 777  18
summary(College) ### Resumo das variáveis
##  Private        Apps           Accept          Enroll       Top10perc    
##  No :212   Min.   :   81   Min.   :   72   Min.   :  35   Min.   : 1.00  
##  Yes:565   1st Qu.:  776   1st Qu.:  604   1st Qu.: 242   1st Qu.:15.00  
##            Median : 1558   Median : 1110   Median : 434   Median :23.00  
##            Mean   : 3002   Mean   : 2019   Mean   : 780   Mean   :27.56  
##            3rd Qu.: 3624   3rd Qu.: 2424   3rd Qu.: 902   3rd Qu.:35.00  
##            Max.   :48094   Max.   :26330   Max.   :6392   Max.   :96.00  
##    Top25perc      F.Undergrad     P.Undergrad         Outstate    
##  Min.   :  9.0   Min.   :  139   Min.   :    1.0   Min.   : 2340  
##  1st Qu.: 41.0   1st Qu.:  992   1st Qu.:   95.0   1st Qu.: 7320  
##  Median : 54.0   Median : 1707   Median :  353.0   Median : 9990  
##  Mean   : 55.8   Mean   : 3700   Mean   :  855.3   Mean   :10441  
##  3rd Qu.: 69.0   3rd Qu.: 4005   3rd Qu.:  967.0   3rd Qu.:12925  
##  Max.   :100.0   Max.   :31643   Max.   :21836.0   Max.   :21700  
##    Room.Board       Books           Personal         PhD        
##  Min.   :1780   Min.   :  96.0   Min.   : 250   Min.   :  8.00  
##  1st Qu.:3597   1st Qu.: 470.0   1st Qu.: 850   1st Qu.: 62.00  
##  Median :4200   Median : 500.0   Median :1200   Median : 75.00  
##  Mean   :4358   Mean   : 549.4   Mean   :1341   Mean   : 72.66  
##  3rd Qu.:5050   3rd Qu.: 600.0   3rd Qu.:1700   3rd Qu.: 85.00  
##  Max.   :8124   Max.   :2340.0   Max.   :6800   Max.   :103.00  
##     Terminal       S.F.Ratio      perc.alumni        Expend     
##  Min.   : 24.0   Min.   : 2.50   Min.   : 0.00   Min.   : 3186  
##  1st Qu.: 71.0   1st Qu.:11.50   1st Qu.:13.00   1st Qu.: 6751  
##  Median : 82.0   Median :13.60   Median :21.00   Median : 8377  
##  Mean   : 79.7   Mean   :14.09   Mean   :22.74   Mean   : 9660  
##  3rd Qu.: 92.0   3rd Qu.:16.50   3rd Qu.:31.00   3rd Qu.:10830  
##  Max.   :100.0   Max.   :39.80   Max.   :64.00   Max.   :56233  
##    Grad.Rate     
##  Min.   : 10.00  
##  1st Qu.: 53.00  
##  Median : 65.00  
##  Mean   : 65.46  
##  3rd Qu.: 78.00  
##  Max.   :118.00
### Vamos considerar Grad.Rate (taxa de formados) como a variável resposta na nossa análise. Começamos a análise com alguns gráficos.

ggplot(College, aes(x = Grad.Rate)) + geom_histogram() +
    theme_bw(base_size = 14)
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

### Distribuição das taxas de formados

ggplot(College, aes(x = Top10perc, y = Grad.Rate)) + geom_point() +
    geom_smooth(method = "loess") + 
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

### Taxas de formados versus percentual de alunos entre os 10% melhores
### no ensino médio.

ggplot(College, aes(x = Outstate, y = Grad.Rate)) + geom_point() +
    geom_smooth(method = "loess") + 
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

### Taxas de formados versus investimentos externos.

ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
    geom_smooth(method = "loess") +
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

### Taxas de formados versus porcentagens de ex-alunos contribuintes.
ggpairs(College, proportions = "auto") ### Matriz de gráficos de dispersão. - DEMORA PARA RODAR

#para salvar o ultimo grafico
ggsave("plot_grande.png", device = "png", width = 100, height = 80, units = "cm")
# tamanho maximo 50 in ou 126 cm
Grafico pesado
Grafico pesado
ggcorr(College[,-1], label = TRUE, label_round = 2) ### Correlograma.
ggsave("plot_colorido.png", device = "png", width = 30, height = 20, units = "cm")
Grafico colorido
Grafico colorido
### Parte 1 - Ajuste dos modelos lineares. Comecemos com o caso de apenas uma 
### variável explicativa (no caso, perc.alumni)

### Para ajustar modelos lineares no R usamos a função lm. Vamos consultar
### a documentação da função.

#help('lm')

Multipla (1 var resposta) e Multivarida (mais var respostas) são coisas diferentes

### Ajuste da regressão linear simples (assumindo relação linear entre a taxa de formados e o percentual de ex-alunos contribuintes)

ajuste1 <- lm(Grad.Rate ~ perc.alumni, data = College) ###(resposta ~ var.explicativas)

ajuste1
## 
## Call:
## lm(formula = Grad.Rate ~ perc.alumni, data = College)
## 
## Coefficients:
## (Intercept)  perc.alumni  
##     49.9863       0.6805

\[ \hat{Grad.Rate} = 49,9863 + 0,6805 \times perc.alumni \] chapeu indica estimado?

summary(ajuste1)
## 
## Call:
## lm(formula = Grad.Rate ~ perc.alumni, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -58.247  -9.513   0.043   9.362  54.404 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 49.98633    1.12345   44.49   <2e-16 ***
## perc.alumni  0.68049    0.04338   15.69   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.98 on 775 degrees of freedom
## Multiple R-squared:  0.241,  Adjusted R-squared:   0.24 
## F-statistic: 246.1 on 1 and 775 DF,  p-value: < 2.2e-16
### O percentual de ex-alunos contribuintes tem efeito positivo, e 
### estatisticamente significativo na taxa de formados.
### Vamos visualizar o ajuste do modelo
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
    stat_smooth(method = "lm") +
    theme_bw(base_size = 14)
## `geom_smooth()` using formula = 'y ~ x'

### Vamos investigar possível efeito quadrático do percentual de contribuíntes na taxa de formados. Para isso, adicionamos ao preditor o termo quadrático da variável explicativa, da seguinte forma:

ajuste2 <- lm(Grad.Rate ~ perc.alumni + I(perc.alumni^2), data = College)
summary(ajuste2)
## 
## Call:
## lm(formula = Grad.Rate ~ perc.alumni + I(perc.alumni^2), data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -57.444  -9.146   0.042   9.448  53.446 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      45.896244   1.877789  24.442  < 2e-16 ***
## perc.alumni       1.085948   0.155614   6.978 6.42e-12 ***
## I(perc.alumni^2) -0.007652   0.002821  -2.712  0.00683 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.91 on 774 degrees of freedom
## Multiple R-squared:  0.2481, Adjusted R-squared:  0.2462 
## F-statistic: 127.7 on 2 and 774 DF,  p-value: < 2.2e-16
### O termo quadrático é estatisticamente significativo, indicando que a
### relação entre as variáveis não é linear. Vamos dar um passo além, e
### incluir o termo de terceira ordem para o percentual de contribuintes
### (modelo cúbico).

\[ \hat{Grad.Rate} = 45.896244 + 1.085948 \times perc.alumni -0.007652 \times perc.alumni^2 \]

de acordo com os valores de p é significativo, então manteria essa variavel

ajuste3 <- lm(Grad.Rate ~ perc.alumni + I(perc.alumni^2) + I(perc.alumni^3), data = College)
summary(ajuste3)
## 
## Call:
## lm(formula = Grad.Rate ~ perc.alumni + I(perc.alumni^2) + I(perc.alumni^3), 
##     data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -56.571  -9.123   0.005   9.346  52.896 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      42.9519252  2.8823421  14.902  < 2e-16 ***
## perc.alumni       1.5681680  0.3905839   4.015 6.53e-05 ***
## I(perc.alumni^2) -0.0276219  0.0151030  -1.829   0.0678 .  
## I(perc.alumni^3)  0.0002297  0.0001706   1.346   0.1787    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.91 on 773 degrees of freedom
## Multiple R-squared:  0.2499, Adjusted R-squared:  0.247 
## F-statistic: 85.84 on 3 and 773 DF,  p-value: < 2.2e-16
### O termo de ordem cúbica não tem significância estatística. Vamos seguir
### a análise com o modelo quadrático.

\[ \hat{Grad.Rate} = 42.9519252 + 1.5681680 \times perc.alumni -0.0276219 \times perc.alumni^2 + 0.0002297 \times perc.alumni^3 \] de acordo com os valores de p é significativo, então manteria essa variavel

### Vamos extrair alguns elementos do modelo ajustado
ajuste2$coefficients ### Estimativas dos parâmetros
##      (Intercept)      perc.alumni I(perc.alumni^2) 
##     45.896244242      1.085947860     -0.007651754
ajuste2$residuals ### Resíduos ordeinários
##                  Abilene Christian University 
##                                    2.17423398 
##                            Adelphi University 
##                                   -5.31256105 
##                                Adrian College 
##                                  -17.58810167 
##                           Agnes Scott College 
##                                  -16.60106419 
##                     Alaska Pacific University 
##                                  -33.03753295 
##                             Albertson College 
##                                   -1.91580850 
##                       Albertus Magnus College 
##                                   -5.95830307 
##                                Albion College 
##                                   -2.60106419 
##                              Albright College 
##                                   13.17473271 
##                     Alderson-Broaddus College 
##                                   -8.46381755 
##                             Alfred University 
##                                    0.79270745 
##                             Allegheny College 
##                                   -1.55750846 
##       Allentown Coll. of St. Francis de Sales 
##                                    8.67327410 
##                                  Alma College 
##                                   -4.81117993 
##                               Alverno College 
##                                  -13.95830307 
##                American International College 
##                                    5.23302952 
##                               Amherst College 
##                                   16.05885120 
##                           Anderson University 
##                                   -0.59977055 
##                            Andrews University 
##                                  -16.96413751 
##                       Angelo State University 
##                                  -17.13468970 
##                            Antioch University 
##                                  -26.53102101 
##                  Appalachian State University 
##                                   10.40022945 
##                               Aquinas College 
##                                   -3.26259465 
##          Arizona State University Main campus 
##                                   -3.13468970 
##               Arkansas College (Lyon College) 
##                                  -13.55158273 
##                      Arkansas Tech University 
##                                   -3.13468970 
##                            Assumption College 
##                                   16.41189833 
##                 Auburn University-Main Campus 
##                                    6.03586249 
##                              Augsburg College 
##                                  -14.20729255 
##                          Augustana College IL 
##                                   -6.09135265 
##                             Augustana College 
##                                   -2.58810167 
##                                Austin College 
##                                   -2.39976380 
##                               Averett College 
##                                   -8.91580850 
##                              Baker University 
##                                   -0.32672590 
##                       Baldwin-Wallace College 
##                                   20.44550005 
##                                 Barat College 
##                                   -3.53102101 
##                                  Bard College 
##                                    7.41189833 
##                               Barnard College 
##                                   17.60023620 
##                              Barry University 
##                                   15.08419150 
##                             Baylor University 
##                                   -4.11313051 
##                                Beaver College 
##                                    1.41189833 
##                            Bellarmine College 
##                                   11.79270745 
##                         Belmont Abbey College 
##                                   -3.99054747 
##                            Belmont University 
##                                  -14.76697048 
##                                Beloit College 
##                                    0.04169693 
##                      Bemidji State University 
##                                  -15.31256105 
##                           Benedictine College 
##                                   -4.96413751 
##                            Bennington College 
##                                  -18.39976380 
##                               Bentley College 
##                                   17.44550005 
##                                 Berry College 
##                                    0.85399897 
##                               Bethany College 
##                                  -22.95360728 
##                             Bethel College KS 
##                                  -16.81117993 
##                                Bethel College 
##                                  -23.72042004 
##                       Bethune Cookman College 
##                                    2.95001707 
##                   Birmingham-Southern College 
##                                   -1.97304415 
##                             Blackburn College 
##                                  -30.95770455 
##              Bloomsburg Univ. of Pennsylvania 
##                                   11.23302952 
##                             Bluefield College 
##                                    3.91477796 
##                              Bluffton College 
##                                   -5.76697048 
##                             Boston University 
##                                   10.68743895 
##                               Bowdoin College 
##                                   14.32480917 
##                Bowling Green State University 
##                                    7.40022945 
##                              Bradford College 
##                                  -19.32672590 
##                            Bradley University 
##                                   19.67327410 
##                           Brandeis University 
##                                   16.44841727 
##                             Brenau University 
##                                   -8.82576602 
##                        Brewton-Parker College 
##                                  -37.99054747 
##                           Briar Cliff College 
##                                  -10.95830307 
##                           Bridgewater College 
##                                   14.44841727 
##             Brigham Young University at Provo 
##                                  -44.09135265 
##                              Brown University 
##                                   20.39010666 
##                             Bryn Mawr College 
##                                    8.26417136 
##                           Bucknell University 
##                                   17.92630565 
##                           Buena Vista College 
##                                   22.00945253 
##                             Butler University 
##                                   12.04639272 
##                               Cabrini College 
##                                   -4.07369435 
##                              Caldwell College 
##                                  -13.26259465 
##                California Lutheran University 
##                                   -2.14600103 
##               California Polytechnic-San Luis 
##                                    0.27957996 
##         California State University at Fresno 
##                                    6.90588512 
##                                Calvin College 
##                                    3.44249154 
##                           Campbell University 
##                                  -10.97304415 
##                        Campbellsville College 
##                                   -9.72042004 
##                              Canisius College 
##                                   -4.95830307 
##                            Capital University 
##                                   13.36129202 
##                               Capitol College 
##                                  -15.55158273 
##                              Carleton College 
##                                    7.49319765 
##                    Carnegie Mellon University 
##                                    1.79270745 
##                               Carroll College 
##                                   10.73740535 
##                         Carson-Newman College 
##                                    5.68743895 
##                              Carthage College 
##                                   -4.08364835 
##               Case Western Reserve University 
##                                   -3.95360728 
##                       Castleton State College 
##                                   24.90588512 
##                               Catawba College 
##                                  -14.63870798 
##                Catholic University of America 
##                                   12.03586249 
##                             Cazenovia College 
##                                   53.44550005 
##                           Cedar Crest College 
##                                   -2.60989334 
##                            Cedarville College 
##                                   -9.97304415 
##                             Centenary College 
##                                  -40.55449995 
##                Centenary College of Louisiana 
##                                   -2.26259465 
##                   Center for Creative Studies 
##                                   -3.11760762 
##                               Central College 
##                                   -3.95360728 
##          Central Connecticut State University 
##                                   -1.11760762 
##             Central Missouri State University 
##                                   -0.11760762 
##                 Central Washington University 
##                                    5.10375576 
##                      Central Wesleyan College 
##                                   -4.96413751 
##                                Centre College 
##                                   -9.50680235 
##                            Chapman University 
##                                   -5.13646827 
##                               Chatham College 
##                                  -23.60106419 
##                         Chestnut Hill College 
##                                    2.04639272 
##                           Christendom College 
##                                   -4.14600103 
##                 Christian Brothers University 
##                                   -4.55158273 
##                Christopher Newport University 
##                                  -13.31256105 
##                               Claflin College 
##                                  -51.20729255 
##                     Claremont McKenna College 
##                                    5.32480917 
##                              Clark University 
##                                    4.46897899 
##                                Clarke College 
##                                    4.36129202 
##                           Clarkson University 
##                                    4.18882007 
##                            Clemson University 
##                                   10.85399897 
## Clinch Valley Coll. of  the Univ. of Virginia 
##                                   -9.04998293 
##                                   Coe College 
##                                   -5.81117993 
##                                 Coker College 
##                                   -1.60989334 
##                                 Colby College 
##                                   13.44249154 
##                            Colgate University 
##                                   13.73090339 
##                          College Misericordia 
##                                   29.17473271 
##                         College of Charleston 
##                                  -11.96413751 
##                   College of Mount St. Joseph 
##                                   25.46897899 
##                  College of Mount St. Vincent 
##                                    8.46897899 
##                         College of Notre Dame 
##                                  -10.12294333 
##             College of Notre Dame of Maryland 
##                                    2.18882007 
##                     College of Saint Benedict 
##                                    3.04169693 
##                    College of Saint Catherine 
##                                    4.18882007 
##                    College of Saint Elizabeth 
##                                   11.17473271 
##                         College of Saint Rose 
##                                   -6.30380939 
##                           College of Santa Fe 
##                                   26.87705667 
##                         College of St. Joseph 
##                                   12.23302952 
##                    College of St. Scholastica 
##                                   -1.39976380 
##                     College of the Holy Cross 
##                                   12.52317854 
##                   College of William and Mary 
##                                   20.79270745 
##                            College of Wooster 
##                                   -9.44390955 
##                              Colorado College 
##                                    2.62262639 
##                     Colorado State University 
##                                    3.00945253 
##                           Columbia College MO 
##                                  -11.03753295 
##                              Columbia College 
##                                    2.02695585 
##                           Columbia University 
##                                   33.67327410 
##                 Concordia College at St. Paul 
##                                  -17.96413751 
##                    Concordia Lutheran College 
##                                  -13.04998293 
##                       Concordia University CA 
##                                   -3.72042004 
##                          Concordia University 
##                                   16.27957996 
##                           Connecticut College 
##                                   13.90864735 
##                              Converse College 
##                                    2.79270745 
##                               Cornell College 
##                                   -8.20729255 
##                          Creighton University 
##                                   12.18882007 
##                       Culver-Stockton College 
##                                  -19.30380939 
##                            Cumberland College 
##                                   12.88239238 
##                            D'Youville College 
##                                  -32.00836076 
##                                  Dana College 
##                                  -14.26259465 
##                        Daniel Webster College 
##                                   16.00945253 
##                             Dartmouth College 
##                                   17.26417136 
##                              Davidson College 
##                                   14.34126512 
##                              Defiance College 
##                                   -7.76697048 
##                        Delta State University 
##                                   -3.31256105 
##                            Denison University 
##                                    1.73090339 
##                             DePauw University 
##                                    9.79270745 
##                             Dickinson College 
##                                   10.39010666 
##                    Dickinson State University 
##                                  -24.30380939 
##                            Dillard University 
##                                    3.17423398 
##                                 Doane College 
##                                  -18.00836076 
##                 Dominican College of Blauvelt 
##                                    2.86531030 
##                                 Dordt College 
##                                   24.85399897 
##                               Dowling College 
##                                  -11.12294333 
##                              Drake University 
##                                    9.44841727 
##                               Drew University 
##                                   12.69619061 
##                                 Drury College 
##                                   -7.53102101 
##                               Duke University 
##                                   18.13584517 
##                               Earlham College 
##                                   -1.65873488 
##                      East Carolina University 
##                                   -4.96413751 
##               East Tennessee State University 
##                                  -13.04998293 
##                 East Texas Baptist University 
##                                  -15.12294333 
##                               Eastern College 
##                                  -12.08364835 
##          Eastern Connecticut State University 
##                                   -9.59977055 
##                   Eastern Illinois University 
##                                   24.86531030 
##                     Eastern Mennonite College 
##                                   11.04639272 
##                      Eastern Nazarene College 
##                                    7.85399897 
##                                Eckerd College 
##                                   -9.95830307 
##                         Elizabethtown College 
##                                   12.73740535 
##                                Elmira College 
##                                   -1.32672590 
##                                  Elms College 
##                                   20.67327410 
##                                  Elon College 
##                                  -10.97304415 
##          Embry Riddle Aeronautical University 
##                                   -6.11760762 
##                         Emory & Henry College 
##                                    0.62262639 
##                              Emory University 
##                                   25.69619061 
##                      Emporia State University 
##                                   -0.11760762 
##                               Erskine College 
##                                  -13.03306964 
##                                Eureka College 
##                                  -19.20729255 
##                       Evergreen State College 
##                                    8.40022945 
##                          Fairfield University 
##                                   22.41189833 
##                 Fayetteville State University 
##                                  -31.99054747 
##                                Ferrum College 
##                                  -33.04998293 
##                               Flagler College 
##                                   13.95001707 
##               Florida Institute of Technology 
##                                    3.87705667 
##              Florida International University 
##                                    1.44550005 
##                      Florida Southern College 
##                                   12.00945253 
##                      Florida State University 
##                                   -2.46381755 
##                             Fontbonne College 
##                                   -5.55158273 
##                            Fordham University 
##                                   20.40022945 
##                            Fort Lewis College 
##                                   -6.13646827 
##                     Francis Marion University 
##                                  -11.09411488 
##         Franciscan University of Steubenville 
##                                   28.90588512 
##                              Franklin College 
##                                  -15.60106419 
##                       Franklin Pierce College 
##                                  -10.31256105 
##                     Freed-Hardeman University 
##                                    3.27957996 
##                        Fresno Pacific College 
##                                   -5.59977055 
##                             Furman University 
##                                   11.69619061 
##                             Gannon University 
##                                    2.03586249 
##                       Gardner Webb University 
##                                  -28.82576602 
##                                Geneva College 
##                                    5.04169693 
##                            George Fox College 
##                                  -14.08364835 
##                       George Mason University 
##                                   -7.12294333 
##                  George Washington University 
##                                   11.53618245 
##                            Georgetown College 
##                                  -15.30380939 
##                         Georgetown University 
##                                   25.36129202 
##               Georgia Institute of Technology 
##                                   -3.39976380 
##                      Georgia State University 
##                                  -21.99054747 
##                        Georgian Court College 
##                                    6.36129202 
##                            Gettysburg College 
##                                   10.18882007 
##                         Goldey Beacom College 
##                                  -14.11760762 
##                            Gonzaga University 
##                                   -3.81117993 
##                                Gordon College 
##                                   -6.81117993 
##                                Goshen College 
##                                   -7.65873488 
##                               Goucher College 
##                                    3.02695585 
##                    Grace College and Seminary 
##                                   -4.95830307 
##                             Graceland College 
##                                  -20.55158273 
##                 Grand Valley State University 
##                                    1.95001707 
##                        Green Mountain College 
##                                    8.44841727 
##                            Greensboro College 
##                                  -33.20729255 
##                            Greenville College 
##                                   -1.31256105 
##                              Grinnell College 
##                                    0.77508524 
##                            Grove City College 
##                                   37.03586249 
##                              Guilford College 
##                                   -6.58810167 
##                     Gustavus Adolphus College 
##                                   -3.14072052 
##                         Gwynedd Mercy College 
##                                   29.91635165 
##                              Hamilton College 
##                                    7.49319765 
##                            Hamline University 
##                                   -8.39976380 
##                      Hampden - Sydney College 
##                                  -12.95770455 
##                            Hampton University 
##                                   14.95001707 
##                               Hanover College 
##                                   -4.95830307 
##                     Hardin-Simmons University 
##                                  -17.99054747 
##                            Harding University 
##                                   -2.60106419 
##                              Hartwick College 
##                                    0.18882007 
##                            Harvard University 
##                                   18.32480917 
##                           Harvey Mudd College 
##                                   20.34126512 
##                              Hastings College 
##                                  -10.14600103 
##                               Hendrix College 
##                                   -5.95830307 
##                             Hillsdale College 
##                                    6.79270745 
##                                 Hiram College 
##                                    5.02695585 
##             Hobart and William Smith Colleges 
##                                    3.39893581 
##                            Hofstra University 
##                                    4.00945253 
##                               Hollins College 
##                                   -8.39210089 
##                                  Hood College 
##                                   -1.97304415 
##                                  Hope College 
##                                   -5.09135265 
##                              Houghton College 
##                                   -0.55158273 
##                            Huntingdon College 
##                                  -11.04998293 
##                            Huntington College 
##                                  -21.26259465 
##                              Huron University 
##                                  -29.11760762 
##                                Husson College 
##                                   33.88239238 
##                  Illinois Benedictine College 
##                                    4.04639272 
##                              Illinois College 
##                                  -19.58810167 
##              Illinois Institute of Technology 
##                                  -12.95830307 
##                     Illinois State University 
##                                   -7.31256105 
##                  Illinois Wesleyan University 
##                                    9.02695585 
##                            Immaculata College 
##                                   -4.39976380 
##                        Incarnate Word College 
##                                   29.67327410 
##                      Indiana State University 
##                                  -14.09411488 
##             Indiana University at Bloomington 
##                                    0.44841727 
##                   Indiana Wesleyan University 
##                                  -26.46381755 
##                                  Iona College 
##                                    6.40022945 
##                         Iowa State University 
##                                   -1.08364835 
##                                Ithaca College 
##                                    6.73740535 
##                      James Madison University 
##                                   27.04639272 
##                             Jamestown College 
##                                  -11.32672590 
##                     Jersey City State College 
##                                  -29.99054747 
##                         John Brown University 
##                                   11.23302952 
##                       John Carroll University 
##                                   18.69619061 
##                      Johns Hopkins University 
##                                   13.88686949 
##                         Johnson State College 
##                                  -24.46381755 
##                                Judson College 
##                                  -15.58810167 
##                               Juniata College 
##                                    4.39893581 
##                       Kansas State University 
##                                  -12.08364835 
##                    Kansas Wesleyan University 
##                                    8.40022945 
##                           Keene State College 
##                                    2.27957996 
##                     Kentucky Wesleyan College 
##                                  -10.81117993 
##                                Kenyon College 
##                                    8.34126512 
##                                 Keuka College 
##                                  -27.44390955 
##                                King's College 
##                                   11.39893581 
##                                  King College 
##                                   -3.26259465 
##                                  Knox College 
##                                    5.60023620 
##                              La Roche College 
##                                   -7.59977055 
##                           La Salle University 
##                                   28.95001707 
##                             Lafayette College 
##                                   15.88686949 
##                              LaGrange College 
##                                   17.17423398 
##                           Lake Forest College 
##                                   13.23302952 
##                              Lakeland College 
##                                  -11.26259465 
##                              Lamar University 
##                                  -31.82576602 
##                            Lambuth University 
##                                    4.00945253 
##                             Lander University 
##                                   -5.91580850 
##                           Lawrence University 
##                                   -5.93472434 
##                              Le Moyne College 
##                                   23.69619061 
##                        Lebanon Valley College 
##                                   13.41189833 
##                             Lehigh University 
##                                   12.55609045 
##                          Lenoir-Rhyne College 
##                                    1.44550005 
##                                Lesley College 
##                                   -4.96413751 
##                         LeTourneau University 
##                                  -10.82526729 
##                       Lewis and Clark College 
##                                    3.67327410 
##                              Lewis University 
##                                    5.00945253 
##                   Lincoln Memorial University 
##                                  -21.53102101 
##                            Lincoln University 
##                                   -9.09411488 
##                            Lindenwood College 
##                                   44.95001707 
##                              Linfield College 
##                                    7.02695585 
##                           Livingstone College 
##                                   30.68743895 
##         Lock Haven University of Pennsylvania 
##                                    3.40022945 
##                              Longwood College 
##                                   -4.82526729 
##                                 Loras College 
##                                    2.44841727 
##                             Louisiana College 
##                                    2.08419150 
##     Louisiana State University at Baton Rouge 
##                                  -19.91580850 
##                     Louisiana Tech University 
##                                  -13.72042004 
##                                Loyola College 
##                                   10.36129202 
##                   Loyola Marymount University 
##                                   28.00945253 
##                             Loyola University 
##                                   -6.59977055 
##                     Loyola University Chicago 
##                                    4.53618245 
##                                Luther College 
##                                    0.88686949 
##                              Lycoming College 
##                                   -0.81117993 
##                             Lynchburg College 
##                                    2.44841727 
##                          Lyndon State College 
##                                   -9.46381755 
##                            Macalester College 
##                                    1.39893581 
##                             MacMurray College 
##                                  -14.39976380 
##                                Malone College 
##                                   -6.31256105 
##                            Manchester College 
##                                   -0.55449995 
##                             Manhattan College 
##                                   10.73740535 
##                        Manhattanville College 
##                                    2.44841727 
##                      Mankato State University 
##                                   11.08419150 
##                 Marian College of Fond du Lac 
##                                   -6.32672590 
##                              Marietta College 
##                                   24.41189833 
##                                Marist College 
##                                   -4.97304415 
##                          Marquette University 
##                                    8.73740535 
##                           Marshall University 
##                                   -5.99054747 
##                          Mary Baldwin College 
##                                    8.93574712 
##                       Mary Washington College 
##                                   12.41189833 
##                   Marymount College Tarrytown 
##                                    5.41189833 
##                   Marymount Manhattan College 
##                                    3.44550005 
##                          Marymount University 
##                                   -7.14600103 
##                             Maryville College 
##                                  -27.44390955 
##                          Maryville University 
##                                    2.27957996 
##                              Marywood College 
##                                   -5.58810167 
##         Massachusetts Institute of Technology 
##                                   19.46897899 
##                     Mayville State University 
##                                   -5.91580850 
##                             McKendree College 
##                                  -13.32672590 
##                            McMurry University 
##                                  -24.91580850 
##                             McPherson College 
##                                  -31.26909661 
##                             Mercer University 
##                                   30.53618245 
##                            Mercyhurst College 
##                                    7.04639272 
##                              Meredith College 
##                                    8.60023620 
##                             Merrimack College 
##                                   13.91635165 
##                            Mesa State College 
##                                    1.17423398 
##                               Messiah College 
##                                   17.41189833 
##                    Miami University at Oxford 
##                                   20.44550005 
##                     Michigan State University 
##                                   15.95001707 
##             Michigan Technological University 
##                                   -3.26259465 
##                   MidAmerica Nazarene College 
##                                  -15.55449995 
##              Millersville University of Penn. 
##                                    6.44550005 
##                              Milligan College 
##                                    2.68743895 
##                           Millikin University 
##                                    6.73740535 
##                              Millsaps College 
##                                  -18.11313051 
##               Milwaukee School of Engineering 
##                                    0.17473271 
##                           Mississippi College 
##                                   -1.96413751 
##                  Mississippi State University 
##                                  -11.55449995 
##              Mississippi University for Women 
##                                    8.90588512 
##               Missouri Southern State College 
##                                   44.95001707 
##                       Missouri Valley College 
##                                  -34.31256105 
##                           Monmouth College IL 
##                                  -22.44390955 
##                              Monmouth College 
##                                   -6.46381755 
##       Montana College of Mineral Sci. & Tech. 
##                                    1.79270745 
##                      Montana State University 
##                                  -17.09411488 
##                    Montclair State University 
##                                    2.95001707 
##                     Montreat-Anderson College 
##                                  -35.11760762 
##                     Moorhead State University 
##                                   -3.76697048 
##                              Moravian College 
##                                    3.69619061 
##                             Morehouse College 
##                                   27.00945253 
##                           Morningside College 
##                                  -16.81117993 
##                                Morris College 
##                                  -13.97304415 
##                         Mount Holyoke College 
##                                    2.62262639 
##                           Mount Marty College 
##                                  -32.11313051 
##                            Mount Mary College 
##                                    9.04169693 
##                           Mount Mercy College 
##                                   -7.58810167 
##                     Mount Saint Clare College 
##                                  -57.44390955 
##                    Mount Saint Mary's College 
##                                    4.92630565 
##                      Mount Saint Mary College 
##                                   21.17473271 
##                      Mount St. Mary's College 
##                                    8.23302952 
##                           Mount Union College 
##                                    6.46897899 
##                 Mount Vernon Nazarene College 
##                                    4.87705667 
##                            Muhlenberg College 
##                                    6.39010666 
##                       Murray State University 
##                                  -17.63870798 
##                             Muskingum College 
##                                    3.36129202 
##                     National-Louis University 
##                                   22.96246705 
##                 Nazareth College of Rochester 
##                                   -6.55158273 
##            New Jersey Institute of Technology 
##                                    8.23302952 
##      New Mexico Institute of Mining and Tech. 
##                                  -22.91580850 
##                           New York University 
##                                    9.68743895 
##                              Newberry College 
##                                   10.18882007 
##                            Niagara University 
##                                    0.44550005 
##                     North Adams State College 
##                                   -5.14600103 
##       North Carolina A. & T. State University 
##                                  -11.04998293 
##    North Carolina State University at Raleigh 
##                                   -3.32672590 
##               North Carolina Wesleyan College 
##                                   -4.91580850 
##                         North Central College 
##                                    2.60023620 
##                 North Dakota State University 
##                                  -25.55158273 
##                            North Park College 
##                                    6.44841727 
##           Northeast Missouri State University 
##                                   17.27957996 
##                       Northeastern University 
##                                  -16.14600103 
##                   Northern Arizona University 
##                                  -12.12294333 
##                  Northern Illinois University 
##                                   -0.91580850 
##           Northwest Missouri State University 
##                                  -12.82526729 
##                    Northwest Nazarene College 
##                                   -6.55449995 
##                          Northwestern College 
##                                   -5.97304415 
##                       Northwestern University 
##                                   23.73740535 
##                            Norwich University 
##                                   -3.08364835 
##                            Notre Dame College 
##                                  -35.95830307 
##                            Oakland University 
##                                   -5.72042004 
##                               Oberlin College 
##                                    2.96693036 
##                            Occidental College 
##                                    7.41189833 
##                         Oglethorpe University 
##                                   -2.63870798 
##                      Ohio Northern University 
##                                   10.79270745 
##                               Ohio University 
##                                    5.27957996 
##                      Ohio Wesleyan University 
##                                    2.18882007 
##                   Oklahoma Baptist University 
##                                  -12.96413751 
##                 Oklahoma Christian University 
##                                   -9.09411488 
##                     Oklahoma State University 
##                                  -11.59977055 
##                             Otterbein College 
##                                   15.41189833 
##                   Ouachita Baptist University 
##                                    9.00945253 
##               Our Lady of the Lake University 
##                                  -31.26259465 
##                               Pace University 
##                                    6.00945253 
##                   Pacific Lutheran University 
##                                   16.17473271 
##                         Pacific Union College 
##                                   11.17423398 
##                            Pacific University 
##                                  -24.08364835 
##                     Pembroke State University 
##                                   -3.13468970 
##          Pennsylvania State Univ. Main Campus 
##                                   -0.76697048 
##                         Pepperdine University 
##                                    7.27957996 
##                            Peru State College 
##                                  -23.55158273 
##                              Pfeiffer College 
##                                  -10.72042004 
##       Philadelphia Coll. of Textiles and Sci. 
##                                    7.53618245 
##                           Phillips University 
##                                  -24.76697048 
##                              Piedmont College 
##                                  -31.14600103 
##                             Pikeville College 
##                                    1.40022945 
##                                Pitzer College 
##                                   16.08419150 
##                   Point Loma Nazarene College 
##                                   -9.76697048 
##                            Point Park College 
##                                   10.00945253 
##                        Polytechnic University 
##                                    2.40022945 
##             Prairie View A. and M. University 
##                                  -11.97454035 
##                          Presbyterian College 
##                                    6.99163924 
##                          Princeton University 
##                                   16.77508524 
##                            Providence College 
##                                   21.46897899 
##           Purdue University at West Lafayette 
##                                    6.53618245 
##                                Queens College 
##                                  -17.07369435 
##                             Quincy University 
##                                    7.18882007 
##                            Quinnipiac College 
##                                   12.60023620 
##                            Radford University 
##                                    6.95001707 
##                  Ramapo College of New Jersey 
##                                   -7.09411488 
##                        Randolph-Macon College 
##                                   -2.11313051 
##                Randolph-Macon Woman's College 
##                                    0.44841727 
##                                  Reed College 
##                                   -7.60106419 
##                                 Regis College 
##                                    8.39893581 
##              Rensselaer Polytechnic Institute 
##                                    4.67327410 
##                                Rhodes College 
##                                   -3.03306964 
##                              Rider University 
##                                    3.17473271 
##                                 Ripon College 
##                                  -16.73582864 
##                                Rivier College 
##                                   17.23302952 
##                               Roanoke College 
##                                    3.04169693 
##                             Rockhurst College 
##                                   13.67327410 
##                        Rocky Mountain College 
##                                   -1.63870798 
##                     Roger Williams University 
##                                    6.90588512 
##                               Rollins College 
##                                   23.17473271 
##                                Rosary College 
##                                    9.41189833 
##                   Rowan College of New Jersey 
##                                   -1.13646827 
##                      Rutgers at New Brunswick 
##                                   13.23302952 
##            Rutgers State University at Camden 
##                                   -0.82576602 
##            Rutgers State University at Newark 
##                                   -3.31256105 
##                       Sacred Heart University 
##                                   20.68743895 
##                      Saint Ambrose University 
##                                    8.68743895 
##                          Saint Anselm College 
##                                   26.04639272 
##                  Saint Cloud State University 
##                                    2.00945253 
##                      Saint Francis College IN 
##                                   -2.11760762 
##                         Saint Francis College 
##                                    1.44841727 
##                       Saint John's University 
##                                   -6.11313051 
##                     Saint Joseph's College IN 
##                                    3.23302952 
##                        Saint Joseph's College 
##                                   30.90588512 
##                     Saint Joseph's University 
##                                   24.27957996 
##                          Saint Joseph College 
##                                    3.18882007 
##                        Saint Louis University 
##                                    3.23302952 
##                          Saint Mary's College 
##                                   25.79270745 
##             Saint Mary's College of Minnesota 
##                                   14.23302952 
##               Saint Mary-of-the-Woods College 
##                                   -0.60106419 
##                       Saint Michael's College 
##                                   10.02695585 
##                            Saint Olaf College 
##                                   10.79270745 
##                         Saint Peter's College 
##                                  -13.08364835 
##                         Saint Vincent College 
##                                   15.79270745 
##                       Saint Xavier University 
##                                   -5.46381755 
##                       Salem-Teikyo University 
##                                   42.95001707 
##                                 Salem College 
##                                  -19.65873488 
##                    Salisbury State University 
##                                   -6.96413751 
##                            Samford University 
##                                   -1.14600103 
##                    San Diego State University 
##                                  -12.12294333 
##                        Santa Clara University 
##                                   36.23302952 
##                        Sarah Lawrence College 
##                                   20.03586249 
##              Savannah Coll. of Art and Design 
##                                  -13.95830307 
##                             Schreiner College 
##                                  -14.82526729 
##                               Scripps College 
##                                   -4.55750846 
##                    Seattle Pacific University 
##                                    1.44550005 
##                            Seattle University 
##                                   27.68743895 
##                         Seton Hall University 
##                                    3.53618245 
##                            Seton Hill College 
##                                   -4.60106419 
##              Shippensburg University of Penn. 
##                                   13.27957996 
##                               Shorter College 
##                                   -4.96413751 
##                                 Siena College 
##                                   21.99163924 
##                         Siena Heights College 
##                                  -15.14600103 
##                               Simmons College 
##                                    5.60023620 
##                               Simpson College 
##                                   -5.07369435 
##                           Sioux Falls College 
##                                   -5.12294333 
##                              Skidmore College 
##                                   10.04639272 
##                                 Smith College 
##                                   11.13584517 
##                 South Dakota State University 
##                                   -3.95360728 
##           Southeast Missouri State University 
##                                   -9.09411488 
##             Southeastern Oklahoma State Univ. 
##                                   -2.04998293 
##                   Southern California College 
##                                  -13.91580850 
##  Southern Illinois University at Edwardsville 
##                                  -11.09411488 
##                 Southern Methodist University 
##                                    9.85399897 
##                  Southwest Baptist University 
##                                   12.27957996 
##           Southwest Missouri State University 
##                                   -0.91580850 
##                    Southwest State University 
##                                  -21.20729255 
##                Southwestern Adventist College 
##                                  -21.82576602 
##                          Southwestern College 
##                                   -5.82576602 
##                       Southwestern University 
##                                   -7.53102101 
##                           Spalding University 
##                                   11.90864735 
##                               Spelman College 
##                                    2.03586249 
##                          Spring Arbor College 
##                                   -6.04998293 
##                    St. Bonaventure University 
##                                    5.18882007 
##                            St. John's College 
##                                   -4.95830307 
##                       St. John Fisher College 
##                                   -4.95360728 
##                       St. Lawrence University 
##                                    8.88686949 
##                          St. Martin's College 
##                                  -14.09411488 
##              St. Mary's College of California 
##                                   15.85399897 
##                St. Mary's College of Maryland 
##                                   -3.82526729 
##          St. Mary's University of San Antonio 
##                                   18.87705667 
##                           St. Norbert College 
##                                   12.92630565 
##                            St. Paul's College 
##                                   -9.09411488 
##                    St. Thomas Aquinas College 
##                                    8.27957996 
##                              Stephens College 
##                                   -3.14600103 
##                            Stetson University 
##                                    5.44841727 
##               Stevens Institute of Technology 
##                                    5.60023620 
##                Stockton College of New Jersey 
##                                   10.87705667 
##                             Stonehill College 
##                                   25.41189833 
##                                SUNY at Albany 
##                                   12.68743895 
##                            SUNY at Binghamton 
##                                   19.53618245 
##                               SUNY at Buffalo 
##                                   -4.46381755 
##                           SUNY at Stony Brook 
##                                    3.87705667 
##                    SUNY College  at Brockport 
##                                  -10.59977055 
##                       SUNY College  at Oswego 
##                                   -2.32672590 
##                       SUNY College at Buffalo 
##                                  -15.82576602 
##                      SUNY College at Cortland 
##                                   -9.14600103 
##                      SUNY College at Fredonia 
##                                   10.00945253 
##                       SUNY College at Geneseo 
##                                    7.73740535 
##                     SUNY College at New Paltz 
##                                   -1.09411488 
##                   SUNY College at Plattsburgh 
##                                    3.68743895 
##                       SUNY College at Potsdam 
##                                   -3.14600103 
##                      SUNY College at Purchase 
##                                   -8.09411488 
##                        Susquehanna University 
##                                   14.39893581 
##                           Sweet Briar College 
##                                  -19.39210089 
##                           Syracuse University 
##                                    8.27957996 
##                                 Tabor College 
##                                   -7.46381755 
##                             Talladega College 
##                                   -7.12294333 
##                             Taylor University 
##                                   25.18882007 
##                    Tennessee Wesleyan College 
##                                  -25.31256105 
##            Texas A&M Univ. at College Station 
##                                   -1.95360728 
##             Texas A&M University at Galveston 
##                                  -18.31256105 
##                    Texas Christian University 
##                                   -2.82526729 
##                        Texas Lutheran College 
##                                  -17.55158273 
##                     Texas Southern University 
##                                  -55.32672590 
##                     Texas Wesleyan University 
##                                  -12.99054747 
##                                   The Citadel 
##                                   21.85399897 
##                                 Thiel College 
##                                   -1.31256105 
##                             Tiffin University 
##                                  -38.09135265 
##                       Transylvania University 
##                                   -7.55750846 
##                         Trenton State College 
##                                   28.86353173 
##                          Tri-State University 
##                                   -2.55158273 
##                            Trinity College CT 
##                                   10.60789911 
##                            Trinity College DC 
##                                   20.39893581 
##                            Trinity College VT 
##                                    9.04169693 
##                            Trinity University 
##                                   28.44550005 
##                             Tulane University 
##                                    8.67327410 
##                              Tusculum College 
##                                  -18.30380939 
##                           Tuskegee University 
##                                   11.87705667 
##                              Union College KY 
##                                    8.95001707 
##                              Union College NY 
##                                    7.26417136 
##                 Univ. of Wisconsin at OshKosh 
##                                   -3.59977055 
##           University of Alabama at Birmingham 
##                                  -28.31256105 
##        University of Arkansas at Fayetteville 
##                                  -16.99054747 
##          University of California at Berkeley 
##                                   22.00945253 
##            University of California at Irvine 
##                                    9.08419150 
##                 University of Central Florida 
##                                   -9.04998293 
##                      University of Charleston 
##                                    1.00945253 
##                         University of Chicago 
##                                   14.92630565 
##                      University of Cincinnati 
##                                    1.86353173 
##           University of Connecticut at Storrs 
##                                    9.68743895 
##                          University of Dallas 
##                                   -5.95830307 
##                          University of Dayton 
##                                   24.73740535 
##                        University of Delaware 
##                                   14.53618245 
##                          University of Denver 
##                                    1.67327410 
##                   University of Detroit Mercy 
##                                   -8.59977055 
##                         University of Dubuque 
##                                  -17.96413751 
##                      University of Evansville 
##                                    8.04169693 
##                         University of Florida 
##                                    1.44550005 
##                         University of Georgia 
##                                   -3.08364835 
##                        University of Hartford 
##                                   10.95001707 
##                 University of Hawaii at Manoa 
##                                    1.86353173 
##               University of Illinois - Urbana 
##                                   22.27957996 
##             University of Illinois at Chicago 
##                                  -18.13646827 
##                    University of Indianapolis 
##                                    2.17473271 
##                          University of Kansas 
##                                   -5.14600103 
##                        University of La Verne 
##                                  -19.82526729 
##                      University of Louisville 
##                                  -36.55158273 
##             University of Maine at Farmington 
##                                   -9.95830307 
##                University of Maine at Machias 
##                                   13.88239238 
##           University of Maine at Presque Isle 
##                                  -21.91580850 
##    University of Maryland at Baltimore County 
##                                    2.86353173 
##        University of Maryland at College Park 
##                                    5.17423398 
##        University of Massachusetts at Amherst 
##                                    7.53618245 
##      University of Massachusetts at Dartmouth 
##                                   -8.55449995 
##                           University of Miami 
##                                   -3.14600103 
##           University of Michigan at Ann Arbor 
##                                   18.04169693 
##             University of Minnesota at Duluth 
##                                   -3.91580850 
##             University of Minnesota at Morris 
##                                  -10.31256105 
##           University of Minnesota Twin Cities 
##                                  -30.60106419 
##                     University of Mississippi 
##                                   -6.59977055 
##            University of Missouri at Columbia 
##                                   -2.46381755 
##               University of Missouri at Rolla 
##                                  -17.82526729 
##         University of Missouri at Saint Louis 
##                                  -12.46381755 
##                          University of Mobile 
##                                    1.88239238 
##                      University of Montevallo 
##                                   -3.09411488 
##             University of Nebraska at Lincoln 
##                                  -27.39210089 
##                     University of New England 
##                                    5.27957996 
##                   University of New Hampshire 
##                                   13.68743895 
##     University of North Carolina at Asheville 
##                                  -19.91580850 
##   University of North Carolina at Chapel Hill 
##                                   16.17473271 
##     University of North Carolina at Charlotte 
##                                    8.87705667 
##    University of North Carolina at Greensboro 
##                                   -9.14600103 
##    University of North Carolina at Wilmington 
##                                   -5.46381755 
##                    University of North Dakota 
##                                  -12.31256105 
##                   University of North Florida 
##                                  -12.59977055 
##                     University of North Texas 
##                                  -17.13646827 
##               University of Northern Colorado 
##                                  -14.09411488 
##                   University of Northern Iowa 
##                                    8.04169693 
##                      University of Notre Dame 
##                                   17.34126512 
##                        University of Oklahoma 
##                                  -12.91580850 
##                          University of Oregon 
##                                   -4.72042004 
##                    University of Pennsylvania 
##                                   16.88686949 
##          University of Pittsburgh-Main Campus 
##                                   10.00945253 
##                        University of Portland 
##                                    9.85399897 
##                     University of Puget Sound 
##                                    0.85399897 
##                    University of Rhode Island 
##                                    9.87705667 
##                        University of Richmond 
##                                   27.18882007 
##                       University of Rochester 
##                                   13.17473271 
##                       University of San Diego 
##                                    7.27957996 
##                   University of San Francisco 
##                                    7.90588512 
##       University of Sci. and Arts of Oklahoma 
##                                   -6.08522204 
##                        University of Scranton 
##                                   14.44249154 
##         University of South Carolina at Aiken 
##                                   -1.08522204 
##      University of South Carolina at Columbia 
##                                    0.03586249 
##                   University of South Florida 
##                                   -6.12294333 
##             University of Southern California 
##                                   12.00945253 
##               University of Southern Colorado 
##                                   -9.89624424 
##                University of Southern Indiana 
##                                  -27.32672590 
##            University of Southern Mississippi 
##                                  -21.82526729 
##                   University of St. Thomas MN 
##                                   30.27957996 
##                   University of St. Thomas TX 
##                                   -0.14600103 
##          University of Tennessee at Knoxville 
##                                  -13.08364835 
##              University of Texas at Arlington 
##                                  -21.11760762 
##                 University of Texas at Austin 
##                                    8.08419150 
##            University of Texas at San Antonio 
##                                    0.91477796 
##                        University of the Arts 
##                                    1.95001707 
##                     University of the Pacific 
##                                    5.40022945 
##                       University of the South 
##                                    0.32480917 
##                           University of Tulsa 
##                                   -8.99054747 
##                            University of Utah 
##                                  -18.04998293 
##                         University of Vermont 
##                                   23.00945253 
##                        University of Virginia 
##                                   28.91635165 
##                      University of Washington 
##                                    9.00945253 
##                    University of West Florida 
##                                   -4.82576602 
##                 University of Wisconsin-Stout 
##                                    2.85399897 
##              University of Wisconsin-Superior 
##                                  -24.46381755 
##            University of Wisconsin-Whitewater 
##                                    5.68743895 
##          University of Wisconsin at Green Bay 
##                                   -0.97454035 
##            University of Wisconsin at Madison 
##                                    7.44550005 
##          University of Wisconsin at Milwaukee 
##                                  -16.09411488 
##                         University of Wyoming 
##                                  -13.72042004 
##                         Upper Iowa University 
##                                   14.23302952 
##                               Ursinus College 
##                                    1.90864735 
##                              Ursuline College 
##                                    7.53618245 
##                  Valley City State University 
##                                  -28.26259465 
##                         Valparaiso University 
##                                   28.17473271 
##                         Vanderbilt University 
##                                   14.04169693 
##                                Vassar College 
##                                   13.39010666 
##                          Villanova University 
##                                   28.44841727 
##              Virginia Commonwealth University 
##                                  -11.91580850 
##                     Virginia State University 
##                                  -25.91580850 
##                                 Virginia Tech 
##                                    8.44550005 
##                     Virginia Union University 
##                                  -24.09411488 
##                     Virginia Wesleyan College 
##                                    8.40022945 
##                               Viterbo College 
##                                    0.79270745 
##                              Voorhees College 
##                                    8.91477796 
##                                Wabash College 
##                                  -10.47682146 
##                                Wagner College 
##                                    8.17473271 
##                        Wake Forest University 
##                                   13.39893581 
##                              Walsh University 
##                                   -5.39976380 
##                         Warren Wilson College 
##                                   -1.55449995 
##                              Wartburg College 
##                                   -8.60106419 
##              Washington and Jefferson College 
##                                    8.90864735 
##                 Washington and Lee University 
##                                   10.73090339 
##                            Washington College 
##                                  -10.60106419 
##                   Washington State University 
##                                  -15.58810167 
##                         Washington University 
##                                   17.79270745 
##                           Wayne State College 
##                                  -18.95360728 
##                            Waynesburg College 
##                                   -5.95830307 
##                                Webber College 
##                                   39.88239238 
##                            Webster University 
##                                  -11.59977055 
##                             Wellesley College 
##                                    9.62262639 
##                                 Wells College 
##                                   -9.00836076 
##             Wentworth Institute of Technology 
##                                    9.90588512 
##                                Wesley College 
##                                   23.53618245 
##                           Wesleyan University 
##                                   15.39010666 
##              West Chester University of Penn. 
##                                   -9.31256105 
##                    West Liberty State College 
##                                    4.00945253 
##                West Virginia Wesleyan College 
##                                  -11.00836076 
##                   Western Carolina University 
##                                   -0.04998293 
##                      Western Maryland College 
##                                  -16.60989334 
##                   Western Michigan University 
##                                   -1.91580850 
##                   Western New England College 
##                                   -1.46381755 
##             Western State College of Colorado 
##                                    1.88239238 
##                 Western Washington University 
##                                    5.00945253 
##                       Westfield State College 
##                                    0.44550005 
##                        Westminster College MO 
##                                   -2.55449995 
##                           Westminster College 
##                                    2.44249154 
##         Westminster College of Salt Lake City 
##                                  -23.97304415 
##                              Westmont College 
##                                   24.85399897 
##                            Wheaton College IL 
##                                    7.90864735 
##                        Westminster College PA 
##                                   -6.55750846 
##                       Wheeling Jesuit College 
##                                    2.36129202 
##                               Whitman College 
##                                   -9.37737361 
##                              Whittier College 
##                                  -18.95360728 
##                             Whitworth College 
##                                   15.44550005 
##                            Widener University 
##                                   -0.76697048 
##                             Wilkes University 
##                                   -0.55158273 
##                         Willamette University 
##                                   -7.60106419 
##                        William Jewell College 
##                                   -4.76697048 
##                      William Woods University 
##                                   -9.31256105 
##                              Williams College 
##                                   14.94467606 
##                                Wilson College 
##                                  -11.44390955 
##                               Wingate College 
##                                   36.90588512 
##                       Winona State University 
##                                   -4.96413751 
##                           Winthrop University 
##                                   -9.95830307 
##                    Wisconsin Lutheran College 
##                                  -18.95830307 
##                         Wittenberg University 
##                                    7.04639272 
##                               Wofford College 
##                                   -3.00836076 
##               Worcester Polytechnic Institute 
##                                    8.02695585 
##                       Worcester State College 
##                                  -19.59977055 
##                             Xavier University 
##                                   10.79270745 
##                Xavier University of Louisiana 
##                                  -15.55449995 
##                               Yale University 
##                                   18.26417136 
##                  York College of Pennsylvania 
##                                   28.69619061
ajuste2$fitted.values ### Valores ajustados pelo modelo
##                  Abilene Christian University 
##                                      57.82577 
##                            Adelphi University 
##                                      61.31256 
##                                Adrian College 
##                                      71.58810 
##                           Agnes Scott College 
##                                      75.60106 
##                     Alaska Pacific University 
##                                      48.03753 
##                             Albertson College 
##                                      56.91581 
##                       Albertus Magnus College 
##                                      68.95830 
##                                Albion College 
##                                      75.60106 
##                              Albright College 
##                                      66.82527 
##                     Alderson-Broaddus College 
##                                      60.46382 
##                             Alfred University 
##                                      72.20729 
##                             Allegheny College 
##                                      77.55751 
##       Allentown Coll. of St. Francis de Sales 
##                                      65.32673 
##                                  Alma College 
##                                      72.81118 
##                               Alverno College 
##                                      68.95830 
##                American International College 
##                                      63.76697 
##                               Amherst College 
##                                      83.94115 
##                           Anderson University 
##                                      59.59977 
##                            Andrews University 
##                                      62.96414 
##                       Angelo State University 
##                                      51.13469 
##                            Antioch University 
##                                      74.53102 
##                  Appalachian State University 
##                                      59.59977 
##                               Aquinas College 
##                                      68.26259 
##          Arizona State University Main campus 
##                                      51.13469 
##               Arkansas College (Lyon College) 
##                                      67.55158 
##                      Arkansas Tech University 
##                                      51.13469 
##                            Assumption College 
##                                      71.58810 
##                 Auburn University-Main Campus 
##                                      62.96414 
##                              Augsburg College 
##                                      72.20729 
##                          Augustana College IL 
##                                      77.09135 
##                             Augustana College 
##                                      71.58810 
##                                Austin College 
##                                      73.39976 
##                               Averett College 
##                                      56.91581 
##                              Baker University 
##                                      65.32673 
##                       Baldwin-Wallace College 
##                                      64.55450 
##                                 Barat College 
##                                      74.53102 
##                                  Bard College 
##                                      71.58810 
##                               Barnard College 
##                                      73.39976 
##                              Barry University 
##                                      56.91581 
##                             Baylor University 
##                                      76.11313 
##                                Beaver College 
##                                      71.58810 
##                            Bellarmine College 
##                                      72.20729 
##                         Belmont Abbey College 
##                                      55.99055 
##                            Belmont University 
##                                      63.76697 
##                                Beloit College 
##                                      68.95830 
##                      Bemidji State University 
##                                      61.31256 
##                           Benedictine College 
##                                      62.96414 
##                            Bennington College 
##                                      73.39976 
##                               Bentley College 
##                                      64.55450 
##                                 Berry College 
##                                      62.14600 
##                               Bethany College 
##                                      70.95361 
##                             Bethel College KS 
##                                      72.81118 
##                                Bethel College 
##                                      58.72042 
##                       Bethune Cookman College 
##                                      55.04998 
##                   Birmingham-Southern College 
##                                      73.97304 
##                             Blackburn College 
##                                      81.95770 
##              Bloomsburg Univ. of Pennsylvania 
##                                      63.76697 
##                             Bluefield College 
##                                      49.08522 
##                              Bluffton College 
##                                      63.76697 
##                             Boston University 
##                                      61.31256 
##                               Bowdoin College 
##                                      81.67519 
##                Bowling Green State University 
##                                      59.59977 
##                              Bradford College 
##                                      65.32673 
##                            Bradley University 
##                                      65.32673 
##                           Brandeis University 
##                                      67.55158 
##                             Brenau University 
##                                      57.82577 
##                        Brewton-Parker College 
##                                      55.99055 
##                           Briar Cliff College 
##                                      68.95830 
##                           Bridgewater College 
##                                      67.55158 
##             Brigham Young University at Provo 
##                                      77.09135 
##                              Brown University 
##                                      76.60989 
##                             Bryn Mawr College 
##                                      80.73583 
##                           Bucknell University 
##                                      75.07369 
##                           Buena Vista College 
##                                      55.99055 
##                             Butler University 
##                                      70.95361 
##                               Cabrini College 
##                                      75.07369 
##                              Caldwell College 
##                                      68.26259 
##                California Lutheran University 
##                                      62.14600 
##               California Polytechnic-San Luis 
##                                      58.72042 
##         California State University at Fresno 
##                                      54.09411 
##                                Calvin College 
##                                      77.55751 
##                           Campbell University 
##                                      73.97304 
##                        Campbellsville College 
##                                      58.72042 
##                              Canisius College 
##                                      68.95830 
##                            Capital University 
##                                      69.63871 
##                               Capitol College 
##                                      67.55158 
##                              Carleton College 
##                                      83.50680 
##                    Carnegie Mellon University 
##                                      72.20729 
##                               Carroll College 
##                                      68.26259 
##                         Carson-Newman College 
##                                      61.31256 
##                              Carthage College 
##                                      66.08365 
##               Case Western Reserve University 
##                                      70.95361 
##                       Castleton State College 
##                                      54.09411 
##                               Catawba College 
##                                      69.63871 
##                Catholic University of America 
##                                      62.96414 
##                             Cazenovia College 
##                                      64.55450 
##                           Cedar Crest College 
##                                      76.60989 
##                            Cedarville College 
##                                      73.97304 
##                             Centenary College 
##                                      64.55450 
##                Centenary College of Louisiana 
##                                      68.26259 
##                   Center for Creative Studies 
##                                      50.11761 
##                               Central College 
##                                      70.95361 
##          Central Connecticut State University 
##                                      50.11761 
##             Central Missouri State University 
##                                      50.11761 
##                 Central Washington University 
##                                      45.89624 
##                      Central Wesleyan College 
##                                      62.96414 
##                                Centre College 
##                                      83.50680 
##                            Chapman University 
##                                      52.13647 
##                               Chatham College 
##                                      75.60106 
##                         Chestnut Hill College 
##                                      70.95361 
##                           Christendom College 
##                                      62.14600 
##                 Christian Brothers University 
##                                      67.55158 
##                Christopher Newport University 
##                                      61.31256 
##                               Claflin College 
##                                      72.20729 
##                     Claremont McKenna College 
##                                      81.67519 
##                              Clark University 
##                                      74.53102 
##                                Clarke College 
##                                      69.63871 
##                           Clarkson University 
##                                      72.81118 
##                            Clemson University 
##                                      62.14600 
## Clinch Valley Coll. of  the Univ. of Virginia 
##                                      55.04998 
##                                   Coe College 
##                                      72.81118 
##                                 Coker College 
##                                      76.60989 
##                                 Colby College 
##                                      77.55751 
##                            Colgate University 
##                                      79.26910 
##                          College Misericordia 
##                                      66.82527 
##                         College of Charleston 
##                                      62.96414 
##                   College of Mount St. Joseph 
##                                      74.53102 
##                  College of Mount St. Vincent 
##                                      74.53102 
##                         College of Notre Dame 
##                                      53.12294 
##             College of Notre Dame of Maryland 
##                                      72.81118 
##                     College of Saint Benedict 
##                                      68.95830 
##                    College of Saint Catherine 
##                                      72.81118 
##                    College of Saint Elizabeth 
##                                      66.82527 
##                         College of Saint Rose 
##                                      70.30381 
##                           College of Santa Fe 
##                                      53.12294 
##                         College of St. Joseph 
##                                      63.76697 
##                    College of St. Scholastica 
##                                      73.39976 
##                     College of the Holy Cross 
##                                      82.47682 
##                   College of William and Mary 
##                                      72.20729 
##                            College of Wooster 
##                                      78.44391 
##                              Colorado College 
##                                      81.37737 
##                     Colorado State University 
##                                      55.99055 
##                           Columbia College MO 
##                                      48.03753 
##                              Columbia College 
##                                      73.97304 
##                           Columbia University 
##                                      65.32673 
##                 Concordia College at St. Paul 
##                                      62.96414 
##                    Concordia Lutheran College 
##                                      55.04998 
##                       Concordia University CA 
##                                      58.72042 
##                          Concordia University 
##                                      58.72042 
##                           Connecticut College 
##                                      77.09135 
##                              Converse College 
##                                      72.20729 
##                               Cornell College 
##                                      72.20729 
##                          Creighton University 
##                                      72.81118 
##                       Culver-Stockton College 
##                                      70.30381 
##                            Cumberland College 
##                                      50.11761 
##                            D'Youville College 
##                                      78.00836 
##                                  Dana College 
##                                      68.26259 
##                        Daniel Webster College 
##                                      55.99055 
##                             Dartmouth College 
##                                      80.73583 
##                              Davidson College 
##                                      79.65873 
##                              Defiance College 
##                                      63.76697 
##                        Delta State University 
##                                      61.31256 
##                            Denison University 
##                                      79.26910 
##                             DePauw University 
##                                      72.20729 
##                             Dickinson College 
##                                      76.60989 
##                    Dickinson State University 
##                                      70.30381 
##                            Dillard University 
##                                      57.82577 
##                                 Doane College 
##                                      78.00836 
##                 Dominican College of Blauvelt 
##                                      51.13469 
##                                 Dordt College 
##                                      62.14600 
##                               Dowling College 
##                                      53.12294 
##                              Drake University 
##                                      67.55158 
##                               Drew University 
##                                      70.30381 
##                                 Drury College 
##                                      74.53102 
##                               Duke University 
##                                      78.86415 
##                               Earlham College 
##                                      79.65873 
##                      East Carolina University 
##                                      62.96414 
##               East Tennessee State University 
##                                      55.04998 
##                 East Texas Baptist University 
##                                      53.12294 
##                               Eastern College 
##                                      66.08365 
##          Eastern Connecticut State University 
##                                      59.59977 
##                   Eastern Illinois University 
##                                      51.13469 
##                     Eastern Mennonite College 
##                                      70.95361 
##                      Eastern Nazarene College 
##                                      62.14600 
##                                Eckerd College 
##                                      68.95830 
##                         Elizabethtown College 
##                                      68.26259 
##                                Elmira College 
##                                      65.32673 
##                                  Elms College 
##                                      65.32673 
##                                  Elon College 
##                                      73.97304 
##          Embry Riddle Aeronautical University 
##                                      50.11761 
##                         Emory & Henry College 
##                                      81.37737 
##                              Emory University 
##                                      70.30381 
##                      Emporia State University 
##                                      50.11761 
##                               Erskine College 
##                                      80.03307 
##                                Eureka College 
##                                      72.20729 
##                       Evergreen State College 
##                                      59.59977 
##                          Fairfield University 
##                                      71.58810 
##                 Fayetteville State University 
##                                      55.99055 
##                                Ferrum College 
##                                      55.04998 
##                               Flagler College 
##                                      55.04998 
##               Florida Institute of Technology 
##                                      53.12294 
##              Florida International University 
##                                      64.55450 
##                      Florida Southern College 
##                                      55.99055 
##                      Florida State University 
##                                      60.46382 
##                             Fontbonne College 
##                                      67.55158 
##                            Fordham University 
##                                      59.59977 
##                            Fort Lewis College 
##                                      52.13647 
##                     Francis Marion University 
##                                      54.09411 
##         Franciscan University of Steubenville 
##                                      54.09411 
##                              Franklin College 
##                                      75.60106 
##                       Franklin Pierce College 
##                                      61.31256 
##                     Freed-Hardeman University 
##                                      58.72042 
##                        Fresno Pacific College 
##                                      59.59977 
##                             Furman University 
##                                      70.30381 
##                             Gannon University 
##                                      62.96414 
##                       Gardner Webb University 
##                                      57.82577 
##                                Geneva College 
##                                      68.95830 
##                            George Fox College 
##                                      66.08365 
##                       George Mason University 
##                                      53.12294 
##                  George Washington University 
##                                      60.46382 
##                            Georgetown College 
##                                      70.30381 
##                         Georgetown University 
##                                      69.63871 
##               Georgia Institute of Technology 
##                                      73.39976 
##                      Georgia State University 
##                                      55.99055 
##                        Georgian Court College 
##                                      69.63871 
##                            Gettysburg College 
##                                      72.81118 
##                         Goldey Beacom College 
##                                      50.11761 
##                            Gonzaga University 
##                                      72.81118 
##                                Gordon College 
##                                      72.81118 
##                                Goshen College 
##                                      79.65873 
##                               Goucher College 
##                                      73.97304 
##                    Grace College and Seminary 
##                                      68.95830 
##                             Graceland College 
##                                      67.55158 
##                 Grand Valley State University 
##                                      55.04998 
##                        Green Mountain College 
##                                      67.55158 
##                            Greensboro College 
##                                      72.20729 
##                            Greenville College 
##                                      61.31256 
##                              Grinnell College 
##                                      82.22491 
##                            Grove City College 
##                                      62.96414 
##                              Guilford College 
##                                      71.58810 
##                     Gustavus Adolphus College 
##                                      83.14072 
##                         Gwynedd Mercy College 
##                                      66.08365 
##                              Hamilton College 
##                                      83.50680 
##                            Hamline University 
##                                      73.39976 
##                      Hampden - Sydney College 
##                                      81.95770 
##                            Hampton University 
##                                      55.04998 
##                               Hanover College 
##                                      68.95830 
##                     Hardin-Simmons University 
##                                      55.99055 
##                            Harding University 
##                                      75.60106 
##                              Hartwick College 
##                                      72.81118 
##                            Harvard University 
##                                      81.67519 
##                           Harvey Mudd College 
##                                      79.65873 
##                              Hastings College 
##                                      62.14600 
##                               Hendrix College 
##                                      68.95830 
##                             Hillsdale College 
##                                      72.20729 
##                                 Hiram College 
##                                      73.97304 
##             Hobart and William Smith Colleges 
##                                      75.60106 
##                            Hofstra University 
##                                      55.99055 
##                               Hollins College 
##                                      80.39210 
##                                  Hood College 
##                                      73.97304 
##                                  Hope College 
##                                      77.09135 
##                              Houghton College 
##                                      67.55158 
##                            Huntingdon College 
##                                      55.04998 
##                            Huntington College 
##                                      68.26259 
##                              Huron University 
##                                      50.11761 
##                                Husson College 
##                                      50.11761 
##                  Illinois Benedictine College 
##                                      70.95361 
##                              Illinois College 
##                                      71.58810 
##              Illinois Institute of Technology 
##                                      68.95830 
##                     Illinois State University 
##                                      61.31256 
##                  Illinois Wesleyan University 
##                                      73.97304 
##                            Immaculata College 
##                                      73.39976 
##                        Incarnate Word College 
##                                      65.32673 
##                      Indiana State University 
##                                      54.09411 
##             Indiana University at Bloomington 
##                                      67.55158 
##                   Indiana Wesleyan University 
##                                      60.46382 
##                                  Iona College 
##                                      59.59977 
##                         Iowa State University 
##                                      66.08365 
##                                Ithaca College 
##                                      68.26259 
##                      James Madison University 
##                                      70.95361 
##                             Jamestown College 
##                                      65.32673 
##                     Jersey City State College 
##                                      55.99055 
##                         John Brown University 
##                                      63.76697 
##                       John Carroll University 
##                                      70.30381 
##                      Johns Hopkins University 
##                                      76.11313 
##                         Johnson State College 
##                                      60.46382 
##                                Judson College 
##                                      71.58810 
##                               Juniata College 
##                                      75.60106 
##                       Kansas State University 
##                                      66.08365 
##                    Kansas Wesleyan University 
##                                      59.59977 
##                           Keene State College 
##                                      58.72042 
##                     Kentucky Wesleyan College 
##                                      72.81118 
##                                Kenyon College 
##                                      79.65873 
##                                 Keuka College 
##                                      78.44391 
##                                King's College 
##                                      75.60106 
##                                  King College 
##                                      68.26259 
##                                  Knox College 
##                                      73.39976 
##                              La Roche College 
##                                      59.59977 
##                           La Salle University 
##                                      55.04998 
##                             Lafayette College 
##                                      76.11313 
##                              LaGrange College 
##                                      57.82577 
##                           Lake Forest College 
##                                      63.76697 
##                              Lakeland College 
##                                      68.26259 
##                              Lamar University 
##                                      57.82577 
##                            Lambuth University 
##                                      55.99055 
##                             Lander University 
##                                      56.91581 
##                           Lawrence University 
##                                      82.93472 
##                              Le Moyne College 
##                                      70.30381 
##                        Lebanon Valley College 
##                                      71.58810 
##                             Lehigh University 
##                                      78.44391 
##                          Lenoir-Rhyne College 
##                                      64.55450 
##                                Lesley College 
##                                      62.96414 
##                         LeTourneau University 
##                                      66.82527 
##                       Lewis and Clark College 
##                                      65.32673 
##                              Lewis University 
##                                      55.99055 
##                   Lincoln Memorial University 
##                                      74.53102 
##                            Lincoln University 
##                                      54.09411 
##                            Lindenwood College 
##                                      55.04998 
##                              Linfield College 
##                                      73.97304 
##                           Livingstone College 
##                                      61.31256 
##         Lock Haven University of Pennsylvania 
##                                      59.59977 
##                              Longwood College 
##                                      66.82527 
##                                 Loras College 
##                                      67.55158 
##                             Louisiana College 
##                                      56.91581 
##     Louisiana State University at Baton Rouge 
##                                      56.91581 
##                     Louisiana Tech University 
##                                      58.72042 
##                                Loyola College 
##                                      69.63871 
##                   Loyola Marymount University 
##                                      55.99055 
##                             Loyola University 
##                                      59.59977 
##                     Loyola University Chicago 
##                                      60.46382 
##                                Luther College 
##                                      76.11313 
##                              Lycoming College 
##                                      72.81118 
##                             Lynchburg College 
##                                      67.55158 
##                          Lyndon State College 
##                                      60.46382 
##                            Macalester College 
##                                      75.60106 
##                             MacMurray College 
##                                      73.39976 
##                                Malone College 
##                                      61.31256 
##                            Manchester College 
##                                      64.55450 
##                             Manhattan College 
##                                      68.26259 
##                        Manhattanville College 
##                                      67.55158 
##                      Mankato State University 
##                                      56.91581 
##                 Marian College of Fond du Lac 
##                                      65.32673 
##                              Marietta College 
##                                      71.58810 
##                                Marist College 
##                                      73.97304 
##                          Marquette University 
##                                      68.26259 
##                           Marshall University 
##                                      55.99055 
##                          Mary Baldwin College 
##                                      81.06425 
##                       Mary Washington College 
##                                      71.58810 
##                   Marymount College Tarrytown 
##                                      71.58810 
##                   Marymount Manhattan College 
##                                      64.55450 
##                          Marymount University 
##                                      62.14600 
##                             Maryville College 
##                                      78.44391 
##                          Maryville University 
##                                      58.72042 
##                              Marywood College 
##                                      71.58810 
##         Massachusetts Institute of Technology 
##                                      74.53102 
##                     Mayville State University 
##                                      56.91581 
##                             McKendree College 
##                                      65.32673 
##                            McMurry University 
##                                      56.91581 
##                             McPherson College 
##                                      79.26910 
##                             Mercer University 
##                                      60.46382 
##                            Mercyhurst College 
##                                      70.95361 
##                              Meredith College 
##                                      73.39976 
##                             Merrimack College 
##                                      66.08365 
##                            Mesa State College 
##                                      57.82577 
##                               Messiah College 
##                                      71.58810 
##                    Miami University at Oxford 
##                                      64.55450 
##                     Michigan State University 
##                                      55.04998 
##             Michigan Technological University 
##                                      68.26259 
##                   MidAmerica Nazarene College 
##                                      64.55450 
##              Millersville University of Penn. 
##                                      64.55450 
##                              Milligan College 
##                                      61.31256 
##                           Millikin University 
##                                      68.26259 
##                              Millsaps College 
##                                      76.11313 
##               Milwaukee School of Engineering 
##                                      66.82527 
##                           Mississippi College 
##                                      62.96414 
##                  Mississippi State University 
##                                      64.55450 
##              Mississippi University for Women 
##                                      54.09411 
##               Missouri Southern State College 
##                                      55.04998 
##                       Missouri Valley College 
##                                      61.31256 
##                           Monmouth College IL 
##                                      78.44391 
##                              Monmouth College 
##                                      60.46382 
##       Montana College of Mineral Sci. & Tech. 
##                                      72.20729 
##                      Montana State University 
##                                      54.09411 
##                    Montclair State University 
##                                      55.04998 
##                     Montreat-Anderson College 
##                                      50.11761 
##                     Moorhead State University 
##                                      63.76697 
##                              Moravian College 
##                                      70.30381 
##                             Morehouse College 
##                                      55.99055 
##                           Morningside College 
##                                      72.81118 
##                                Morris College 
##                                      73.97304 
##                         Mount Holyoke College 
##                                      81.37737 
##                           Mount Marty College 
##                                      76.11313 
##                            Mount Mary College 
##                                      68.95830 
##                           Mount Mercy College 
##                                      71.58810 
##                     Mount Saint Clare College 
##                                      78.44391 
##                    Mount Saint Mary's College 
##                                      75.07369 
##                      Mount Saint Mary College 
##                                      66.82527 
##                      Mount St. Mary's College 
##                                      63.76697 
##                           Mount Union College 
##                                      74.53102 
##                 Mount Vernon Nazarene College 
##                                      53.12294 
##                            Muhlenberg College 
##                                      76.60989 
##                       Murray State University 
##                                      69.63871 
##                             Muskingum College 
##                                      69.63871 
##                     National-Louis University 
##                                      48.03753 
##                 Nazareth College of Rochester 
##                                      67.55158 
##            New Jersey Institute of Technology 
##                                      63.76697 
##      New Mexico Institute of Mining and Tech. 
##                                      56.91581 
##                           New York University 
##                                      61.31256 
##                              Newberry College 
##                                      72.81118 
##                            Niagara University 
##                                      64.55450 
##                     North Adams State College 
##                                      62.14600 
##       North Carolina A. & T. State University 
##                                      55.04998 
##    North Carolina State University at Raleigh 
##                                      65.32673 
##               North Carolina Wesleyan College 
##                                      56.91581 
##                         North Central College 
##                                      73.39976 
##                 North Dakota State University 
##                                      67.55158 
##                            North Park College 
##                                      67.55158 
##           Northeast Missouri State University 
##                                      58.72042 
##                       Northeastern University 
##                                      62.14600 
##                   Northern Arizona University 
##                                      53.12294 
##                  Northern Illinois University 
##                                      56.91581 
##           Northwest Missouri State University 
##                                      66.82527 
##                    Northwest Nazarene College 
##                                      64.55450 
##                          Northwestern College 
##                                      73.97304 
##                       Northwestern University 
##                                      68.26259 
##                            Norwich University 
##                                      66.08365 
##                            Notre Dame College 
##                                      68.95830 
##                            Oakland University 
##                                      58.72042 
##                               Oberlin College 
##                                      80.03307 
##                            Occidental College 
##                                      71.58810 
##                         Oglethorpe University 
##                                      69.63871 
##                      Ohio Northern University 
##                                      72.20729 
##                               Ohio University 
##                                      58.72042 
##                      Ohio Wesleyan University 
##                                      72.81118 
##                   Oklahoma Baptist University 
##                                      62.96414 
##                 Oklahoma Christian University 
##                                      54.09411 
##                     Oklahoma State University 
##                                      59.59977 
##                             Otterbein College 
##                                      71.58810 
##                   Ouachita Baptist University 
##                                      55.99055 
##               Our Lady of the Lake University 
##                                      68.26259 
##                               Pace University 
##                                      55.99055 
##                   Pacific Lutheran University 
##                                      66.82527 
##                         Pacific Union College 
##                                      57.82577 
##                            Pacific University 
##                                      66.08365 
##                     Pembroke State University 
##                                      51.13469 
##          Pennsylvania State Univ. Main Campus 
##                                      63.76697 
##                         Pepperdine University 
##                                      58.72042 
##                            Peru State College 
##                                      67.55158 
##                              Pfeiffer College 
##                                      58.72042 
##       Philadelphia Coll. of Textiles and Sci. 
##                                      60.46382 
##                           Phillips University 
##                                      63.76697 
##                              Piedmont College 
##                                      62.14600 
##                             Pikeville College 
##                                      59.59977 
##                                Pitzer College 
##                                      56.91581 
##                   Point Loma Nazarene College 
##                                      63.76697 
##                            Point Park College 
##                                      55.99055 
##                        Polytechnic University 
##                                      59.59977 
##             Prairie View A. and M. University 
##                                      46.97454 
##                          Presbyterian College 
##                                      78.00836 
##                          Princeton University 
##                                      82.22491 
##                            Providence College 
##                                      74.53102 
##           Purdue University at West Lafayette 
##                                      60.46382 
##                                Queens College 
##                                      75.07369 
##                             Quincy University 
##                                      72.81118 
##                            Quinnipiac College 
##                                      73.39976 
##                            Radford University 
##                                      55.04998 
##                  Ramapo College of New Jersey 
##                                      54.09411 
##                        Randolph-Macon College 
##                                      76.11313 
##                Randolph-Macon Woman's College 
##                                      67.55158 
##                                  Reed College 
##                                      75.60106 
##                                 Regis College 
##                                      75.60106 
##              Rensselaer Polytechnic Institute 
##                                      65.32673 
##                                Rhodes College 
##                                      80.03307 
##                              Rider University 
##                                      66.82527 
##                                 Ripon College 
##                                      80.73583 
##                                Rivier College 
##                                      63.76697 
##                               Roanoke College 
##                                      68.95830 
##                             Rockhurst College 
##                                      65.32673 
##                        Rocky Mountain College 
##                                      69.63871 
##                     Roger Williams University 
##                                      54.09411 
##                               Rollins College 
##                                      66.82527 
##                                Rosary College 
##                                      71.58810 
##                   Rowan College of New Jersey 
##                                      52.13647 
##                      Rutgers at New Brunswick 
##                                      63.76697 
##            Rutgers State University at Camden 
##                                      57.82577 
##            Rutgers State University at Newark 
##                                      61.31256 
##                       Sacred Heart University 
##                                      61.31256 
##                      Saint Ambrose University 
##                                      61.31256 
##                          Saint Anselm College 
##                                      70.95361 
##                  Saint Cloud State University 
##                                      55.99055 
##                      Saint Francis College IN 
##                                      50.11761 
##                         Saint Francis College 
##                                      67.55158 
##                       Saint John's University 
##                                      76.11313 
##                     Saint Joseph's College IN 
##                                      63.76697 
##                        Saint Joseph's College 
##                                      54.09411 
##                     Saint Joseph's University 
##                                      58.72042 
##                          Saint Joseph College 
##                                      72.81118 
##                        Saint Louis University 
##                                      63.76697 
##                          Saint Mary's College 
##                                      72.20729 
##             Saint Mary's College of Minnesota 
##                                      63.76697 
##               Saint Mary-of-the-Woods College 
##                                      75.60106 
##                       Saint Michael's College 
##                                      73.97304 
##                            Saint Olaf College 
##                                      72.20729 
##                         Saint Peter's College 
##                                      66.08365 
##                         Saint Vincent College 
##                                      72.20729 
##                       Saint Xavier University 
##                                      60.46382 
##                       Salem-Teikyo University 
##                                      55.04998 
##                                 Salem College 
##                                      79.65873 
##                    Salisbury State University 
##                                      62.96414 
##                            Samford University 
##                                      62.14600 
##                    San Diego State University 
##                                      53.12294 
##                        Santa Clara University 
##                                      63.76697 
##                        Sarah Lawrence College 
##                                      62.96414 
##              Savannah Coll. of Art and Design 
##                                      68.95830 
##                             Schreiner College 
##                                      66.82527 
##                               Scripps College 
##                                      77.55751 
##                    Seattle Pacific University 
##                                      64.55450 
##                            Seattle University 
##                                      61.31256 
##                         Seton Hall University 
##                                      60.46382 
##                            Seton Hill College 
##                                      75.60106 
##              Shippensburg University of Penn. 
##                                      58.72042 
##                               Shorter College 
##                                      62.96414 
##                                 Siena College 
##                                      78.00836 
##                         Siena Heights College 
##                                      62.14600 
##                               Simmons College 
##                                      73.39976 
##                               Simpson College 
##                                      75.07369 
##                           Sioux Falls College 
##                                      53.12294 
##                              Skidmore College 
##                                      70.95361 
##                                 Smith College 
##                                      78.86415 
##                 South Dakota State University 
##                                      70.95361 
##           Southeast Missouri State University 
##                                      54.09411 
##             Southeastern Oklahoma State Univ. 
##                                      55.04998 
##                   Southern California College 
##                                      56.91581 
##  Southern Illinois University at Edwardsville 
##                                      54.09411 
##                 Southern Methodist University 
##                                      62.14600 
##                  Southwest Baptist University 
##                                      58.72042 
##           Southwest Missouri State University 
##                                      56.91581 
##                    Southwest State University 
##                                      72.20729 
##                Southwestern Adventist College 
##                                      57.82577 
##                          Southwestern College 
##                                      57.82577 
##                       Southwestern University 
##                                      74.53102 
##                           Spalding University 
##                                      77.09135 
##                               Spelman College 
##                                      62.96414 
##                          Spring Arbor College 
##                                      55.04998 
##                    St. Bonaventure University 
##                                      72.81118 
##                            St. John's College 
##                                      68.95830 
##                       St. John Fisher College 
##                                      70.95361 
##                       St. Lawrence University 
##                                      76.11313 
##                          St. Martin's College 
##                                      54.09411 
##              St. Mary's College of California 
##                                      62.14600 
##                St. Mary's College of Maryland 
##                                      66.82527 
##          St. Mary's University of San Antonio 
##                                      53.12294 
##                           St. Norbert College 
##                                      75.07369 
##                            St. Paul's College 
##                                      54.09411 
##                    St. Thomas Aquinas College 
##                                      58.72042 
##                              Stephens College 
##                                      62.14600 
##                            Stetson University 
##                                      67.55158 
##               Stevens Institute of Technology 
##                                      73.39976 
##                Stockton College of New Jersey 
##                                      53.12294 
##                             Stonehill College 
##                                      71.58810 
##                                SUNY at Albany 
##                                      61.31256 
##                            SUNY at Binghamton 
##                                      60.46382 
##                               SUNY at Buffalo 
##                                      60.46382 
##                           SUNY at Stony Brook 
##                                      53.12294 
##                    SUNY College  at Brockport 
##                                      59.59977 
##                       SUNY College  at Oswego 
##                                      65.32673 
##                       SUNY College at Buffalo 
##                                      57.82577 
##                      SUNY College at Cortland 
##                                      62.14600 
##                      SUNY College at Fredonia 
##                                      55.99055 
##                       SUNY College at Geneseo 
##                                      68.26259 
##                     SUNY College at New Paltz 
##                                      54.09411 
##                   SUNY College at Plattsburgh 
##                                      61.31256 
##                       SUNY College at Potsdam 
##                                      62.14600 
##                      SUNY College at Purchase 
##                                      54.09411 
##                        Susquehanna University 
##                                      75.60106 
##                           Sweet Briar College 
##                                      80.39210 
##                           Syracuse University 
##                                      58.72042 
##                                 Tabor College 
##                                      60.46382 
##                             Talladega College 
##                                      53.12294 
##                             Taylor University 
##                                      72.81118 
##                    Tennessee Wesleyan College 
##                                      61.31256 
##            Texas A&M Univ. at College Station 
##                                      70.95361 
##             Texas A&M University at Galveston 
##                                      61.31256 
##                    Texas Christian University 
##                                      66.82527 
##                        Texas Lutheran College 
##                                      67.55158 
##                     Texas Southern University 
##                                      65.32673 
##                     Texas Wesleyan University 
##                                      55.99055 
##                                   The Citadel 
##                                      62.14600 
##                                 Thiel College 
##                                      61.31256 
##                             Tiffin University 
##                                      77.09135 
##                       Transylvania University 
##                                      77.55751 
##                         Trenton State College 
##                                      52.13647 
##                          Tri-State University 
##                                      67.55158 
##                            Trinity College CT 
##                                      80.39210 
##                            Trinity College DC 
##                                      75.60106 
##                            Trinity College VT 
##                                      68.95830 
##                            Trinity University 
##                                      64.55450 
##                             Tulane University 
##                                      65.32673 
##                              Tusculum College 
##                                      70.30381 
##                           Tuskegee University 
##                                      53.12294 
##                              Union College KY 
##                                      55.04998 
##                              Union College NY 
##                                      80.73583 
##                 Univ. of Wisconsin at OshKosh 
##                                      59.59977 
##           University of Alabama at Birmingham 
##                                      61.31256 
##        University of Arkansas at Fayetteville 
##                                      55.99055 
##          University of California at Berkeley 
##                                      55.99055 
##            University of California at Irvine 
##                                      56.91581 
##                 University of Central Florida 
##                                      55.04998 
##                      University of Charleston 
##                                      55.99055 
##                         University of Chicago 
##                                      75.07369 
##                      University of Cincinnati 
##                                      52.13647 
##           University of Connecticut at Storrs 
##                                      61.31256 
##                          University of Dallas 
##                                      68.95830 
##                          University of Dayton 
##                                      68.26259 
##                        University of Delaware 
##                                      60.46382 
##                          University of Denver 
##                                      65.32673 
##                   University of Detroit Mercy 
##                                      59.59977 
##                         University of Dubuque 
##                                      62.96414 
##                      University of Evansville 
##                                      68.95830 
##                         University of Florida 
##                                      64.55450 
##                         University of Georgia 
##                                      66.08365 
##                        University of Hartford 
##                                      55.04998 
##                 University of Hawaii at Manoa 
##                                      52.13647 
##               University of Illinois - Urbana 
##                                      58.72042 
##             University of Illinois at Chicago 
##                                      52.13647 
##                    University of Indianapolis 
##                                      66.82527 
##                          University of Kansas 
##                                      62.14600 
##                        University of La Verne 
##                                      66.82527 
##                      University of Louisville 
##                                      67.55158 
##             University of Maine at Farmington 
##                                      68.95830 
##                University of Maine at Machias 
##                                      50.11761 
##           University of Maine at Presque Isle 
##                                      56.91581 
##    University of Maryland at Baltimore County 
##                                      52.13647 
##        University of Maryland at College Park 
##                                      57.82577 
##        University of Massachusetts at Amherst 
##                                      60.46382 
##      University of Massachusetts at Dartmouth 
##                                      64.55450 
##                           University of Miami 
##                                      62.14600 
##           University of Michigan at Ann Arbor 
##                                      68.95830 
##             University of Minnesota at Duluth 
##                                      56.91581 
##             University of Minnesota at Morris 
##                                      61.31256 
##           University of Minnesota Twin Cities 
##                                      75.60106 
##                     University of Mississippi 
##                                      59.59977 
##            University of Missouri at Columbia 
##                                      60.46382 
##               University of Missouri at Rolla 
##                                      66.82527 
##         University of Missouri at Saint Louis 
##                                      60.46382 
##                          University of Mobile 
##                                      50.11761 
##                      University of Montevallo 
##                                      54.09411 
##             University of Nebraska at Lincoln 
##                                      80.39210 
##                     University of New England 
##                                      58.72042 
##                   University of New Hampshire 
##                                      61.31256 
##     University of North Carolina at Asheville 
##                                      56.91581 
##   University of North Carolina at Chapel Hill 
##                                      66.82527 
##     University of North Carolina at Charlotte 
##                                      53.12294 
##    University of North Carolina at Greensboro 
##                                      62.14600 
##    University of North Carolina at Wilmington 
##                                      60.46382 
##                    University of North Dakota 
##                                      61.31256 
##                   University of North Florida 
##                                      59.59977 
##                     University of North Texas 
##                                      52.13647 
##               University of Northern Colorado 
##                                      54.09411 
##                   University of Northern Iowa 
##                                      68.95830 
##                      University of Notre Dame 
##                                      79.65873 
##                        University of Oklahoma 
##                                      56.91581 
##                          University of Oregon 
##                                      58.72042 
##                    University of Pennsylvania 
##                                      76.11313 
##          University of Pittsburgh-Main Campus 
##                                      55.99055 
##                        University of Portland 
##                                      62.14600 
##                     University of Puget Sound 
##                                      62.14600 
##                    University of Rhode Island 
##                                      53.12294 
##                        University of Richmond 
##                                      72.81118 
##                       University of Rochester 
##                                      66.82527 
##                       University of San Diego 
##                                      58.72042 
##                   University of San Francisco 
##                                      54.09411 
##       University of Sci. and Arts of Oklahoma 
##                                      49.08522 
##                        University of Scranton 
##                                      77.55751 
##         University of South Carolina at Aiken 
##                                      49.08522 
##      University of South Carolina at Columbia 
##                                      62.96414 
##                   University of South Florida 
##                                      53.12294 
##             University of Southern California 
##                                      55.99055 
##               University of Southern Colorado 
##                                      45.89624 
##                University of Southern Indiana 
##                                      65.32673 
##            University of Southern Mississippi 
##                                      66.82527 
##                   University of St. Thomas MN 
##                                      58.72042 
##                   University of St. Thomas TX 
##                                      62.14600 
##          University of Tennessee at Knoxville 
##                                      66.08365 
##              University of Texas at Arlington 
##                                      50.11761 
##                 University of Texas at Austin 
##                                      56.91581 
##            University of Texas at San Antonio 
##                                      49.08522 
##                        University of the Arts 
##                                      55.04998 
##                     University of the Pacific 
##                                      59.59977 
##                       University of the South 
##                                      81.67519 
##                           University of Tulsa 
##                                      55.99055 
##                            University of Utah 
##                                      55.04998 
##                         University of Vermont 
##                                      55.99055 
##                        University of Virginia 
##                                      66.08365 
##                      University of Washington 
##                                      55.99055 
##                    University of West Florida 
##                                      57.82577 
##                 University of Wisconsin-Stout 
##                                      62.14600 
##              University of Wisconsin-Superior 
##                                      60.46382 
##            University of Wisconsin-Whitewater 
##                                      61.31256 
##          University of Wisconsin at Green Bay 
##                                      46.97454 
##            University of Wisconsin at Madison 
##                                      64.55450 
##          University of Wisconsin at Milwaukee 
##                                      54.09411 
##                         University of Wyoming 
##                                      58.72042 
##                         Upper Iowa University 
##                                      63.76697 
##                               Ursinus College 
##                                      77.09135 
##                              Ursuline College 
##                                      60.46382 
##                  Valley City State University 
##                                      68.26259 
##                         Valparaiso University 
##                                      66.82527 
##                         Vanderbilt University 
##                                      68.95830 
##                                Vassar College 
##                                      76.60989 
##                          Villanova University 
##                                      67.55158 
##              Virginia Commonwealth University 
##                                      56.91581 
##                     Virginia State University 
##                                      56.91581 
##                                 Virginia Tech 
##                                      64.55450 
##                     Virginia Union University 
##                                      54.09411 
##                     Virginia Wesleyan College 
##                                      59.59977 
##                               Viterbo College 
##                                      72.20729 
##                              Voorhees College 
##                                      49.08522 
##                                Wabash College 
##                                      82.47682 
##                                Wagner College 
##                                      66.82527 
##                        Wake Forest University 
##                                      75.60106 
##                              Walsh University 
##                                      73.39976 
##                         Warren Wilson College 
##                                      64.55450 
##                              Wartburg College 
##                                      75.60106 
##              Washington and Jefferson College 
##                                      77.09135 
##                 Washington and Lee University 
##                                      79.26910 
##                            Washington College 
##                                      75.60106 
##                   Washington State University 
##                                      71.58810 
##                         Washington University 
##                                      72.20729 
##                           Wayne State College 
##                                      70.95361 
##                            Waynesburg College 
##                                      68.95830 
##                                Webber College 
##                                      50.11761 
##                            Webster University 
##                                      59.59977 
##                             Wellesley College 
##                                      81.37737 
##                                 Wells College 
##                                      78.00836 
##             Wentworth Institute of Technology 
##                                      54.09411 
##                                Wesley College 
##                                      60.46382 
##                           Wesleyan University 
##                                      76.60989 
##              West Chester University of Penn. 
##                                      61.31256 
##                    West Liberty State College 
##                                      55.99055 
##                West Virginia Wesleyan College 
##                                      78.00836 
##                   Western Carolina University 
##                                      55.04998 
##                      Western Maryland College 
##                                      76.60989 
##                   Western Michigan University 
##                                      56.91581 
##                   Western New England College 
##                                      60.46382 
##             Western State College of Colorado 
##                                      50.11761 
##                 Western Washington University 
##                                      55.99055 
##                       Westfield State College 
##                                      64.55450 
##                        Westminster College MO 
##                                      64.55450 
##                           Westminster College 
##                                      77.55751 
##         Westminster College of Salt Lake City 
##                                      73.97304 
##                              Westmont College 
##                                      62.14600 
##                            Wheaton College IL 
##                                      77.09135 
##                        Westminster College PA 
##                                      77.55751 
##                       Wheeling Jesuit College 
##                                      69.63871 
##                               Whitman College 
##                                      81.37737 
##                              Whittier College 
##                                      70.95361 
##                             Whitworth College 
##                                      64.55450 
##                            Widener University 
##                                      63.76697 
##                             Wilkes University 
##                                      67.55158 
##                         Willamette University 
##                                      75.60106 
##                        William Jewell College 
##                                      63.76697 
##                      William Woods University 
##                                      61.31256 
##                              Williams College 
##                                      84.05532 
##                                Wilson College 
##                                      78.44391 
##                               Wingate College 
##                                      54.09411 
##                       Winona State University 
##                                      62.96414 
##                           Winthrop University 
##                                      68.95830 
##                    Wisconsin Lutheran College 
##                                      68.95830 
##                         Wittenberg University 
##                                      70.95361 
##                               Wofford College 
##                                      78.00836 
##               Worcester Polytechnic Institute 
##                                      73.97304 
##                       Worcester State College 
##                                      59.59977 
##                             Xavier University 
##                                      72.20729 
##                Xavier University of Louisiana 
##                                      64.55450 
##                               Yale University 
##                                      80.73583 
##                  York College of Pennsylvania 
##                                      70.30381
model.matrix(ajuste2) ### Matriz do modelo (matriz X)
##                                               (Intercept) perc.alumni
## Abilene Christian University                            1          12
## Adelphi University                                      1          16
## Adrian College                                          1          30
## Agnes Scott College                                     1          37
## Alaska Pacific University                               1           2
## Albertson College                                       1          11
## Albertus Magnus College                                 1          26
## Albion College                                          1          37
## Albright College                                        1          23
## Alderson-Broaddus College                               1          15
## Alfred University                                       1          31
## Allegheny College                                       1          41
## Allentown Coll. of St. Francis de Sales                 1          21
## Alma College                                            1          32
## Alverno College                                         1          26
## American International College                          1          19
## Amherst College                                         1          63
## Anderson University                                     1          14
## Andrews University                                      1          18
## Angelo State University                                 1           5
## Antioch University                                      1          35
## Appalachian State University                            1          14
## Aquinas College                                         1          25
## Arizona State University Main campus                    1           5
## Arkansas College (Lyon College)                         1          24
## Arkansas Tech University                                1           5
## Assumption College                                      1          30
## Auburn University-Main Campus                           1          18
## Augsburg College                                        1          31
## Augustana College IL                                    1          40
## Augustana College                                       1          30
## Austin College                                          1          33
## Averett College                                         1          11
## Baker University                                        1          21
## Baldwin-Wallace College                                 1          20
## Barat College                                           1          35
## Bard College                                            1          30
## Barnard College                                         1          33
## Barry University                                        1          11
## Baylor University                                       1          38
## Beaver College                                          1          30
## Bellarmine College                                      1          31
## Belmont Abbey College                                   1          10
## Belmont University                                      1          19
## Beloit College                                          1          26
## Bemidji State University                                1          16
## Benedictine College                                     1          18
## Bennington College                                      1          33
## Bentley College                                         1          20
## Berry College                                           1          17
## Bethany College                                         1          29
## Bethel College KS                                       1          32
## Bethel College                                          1          13
## Bethune Cookman College                                 1           9
## Birmingham-Southern College                             1          34
## Blackburn College                                       1          53
## Bloomsburg Univ. of Pennsylvania                        1          19
## Bluefield College                                       1           3
## Bluffton College                                        1          19
## Boston University                                       1          16
## Bowdoin College                                         1          52
## Bowling Green State University                          1          14
## Bradford College                                        1          21
## Bradley University                                      1          21
## Brandeis University                                     1          24
## Brenau University                                       1          12
## Brewton-Parker College                                  1          10
## Briar Cliff College                                     1          26
## Bridgewater College                                     1          24
## Brigham Young University at Provo                       1          40
## Brown University                                        1          39
## Bryn Mawr College                                       1          49
## Bucknell University                                     1          36
## Buena Vista College                                     1          10
## Butler University                                       1          29
## Cabrini College                                         1          36
## Caldwell College                                        1          25
## California Lutheran University                          1          17
## California Polytechnic-San Luis                         1          13
## California State University at Fresno                   1           8
## Calvin College                                          1          41
## Campbell University                                     1          34
## Campbellsville College                                  1          13
## Canisius College                                        1          26
## Capital University                                      1          27
## Capitol College                                         1          24
## Carleton College                                        1          60
## Carnegie Mellon University                              1          31
## Carroll College                                         1          25
## Carson-Newman College                                   1          16
## Carthage College                                        1          22
## Case Western Reserve University                         1          29
## Castleton State College                                 1           8
## Catawba College                                         1          27
## Catholic University of America                          1          18
## Cazenovia College                                       1          20
## Cedar Crest College                                     1          39
## Cedarville College                                      1          34
## Centenary College                                       1          20
## Centenary College of Louisiana                          1          25
## Center for Creative Studies                             1           4
## Central College                                         1          29
## Central Connecticut State University                    1           4
## Central Missouri State University                       1           4
## Central Washington University                           1           0
## Central Wesleyan College                                1          18
## Centre College                                          1          60
## Chapman University                                      1           6
## Chatham College                                         1          37
## Chestnut Hill College                                   1          29
## Christendom College                                     1          17
## Christian Brothers University                           1          24
## Christopher Newport University                          1          16
## Claflin College                                         1          31
## Claremont McKenna College                               1          52
## Clark University                                        1          35
## Clarke College                                          1          27
## Clarkson University                                     1          32
## Clemson University                                      1          17
## Clinch Valley Coll. of  the Univ. of Virginia           1           9
## Coe College                                             1          32
## Coker College                                           1          39
## Colby College                                           1          41
## Colgate University                                      1          45
## College Misericordia                                    1          23
## College of Charleston                                   1          18
## College of Mount St. Joseph                             1          35
## College of Mount St. Vincent                            1          35
## College of Notre Dame                                   1           7
## College of Notre Dame of Maryland                       1          32
## College of Saint Benedict                               1          26
## College of Saint Catherine                              1          32
## College of Saint Elizabeth                              1          23
## College of Saint Rose                                   1          28
## College of Santa Fe                                     1           7
## College of St. Joseph                                   1          19
## College of St. Scholastica                              1          33
## College of the Holy Cross                               1          55
## College of William and Mary                             1          31
## College of Wooster                                      1          43
## Colorado College                                        1          51
## Colorado State University                               1          10
## Columbia College MO                                     1           2
## Columbia College                                        1          34
## Columbia University                                     1          21
## Concordia College at St. Paul                           1          18
## Concordia Lutheran College                              1           9
## Concordia University CA                                 1          13
## Concordia University                                    1          13
## Connecticut College                                     1          40
## Converse College                                        1          31
## Cornell College                                         1          31
## Creighton University                                    1          32
## Culver-Stockton College                                 1          28
## Cumberland College                                      1           4
## D'Youville College                                      1          42
## Dana College                                            1          25
## Daniel Webster College                                  1          10
## Dartmouth College                                       1          49
## Davidson College                                        1          46
## Defiance College                                        1          19
## Delta State University                                  1          16
## Denison University                                      1          45
## DePauw University                                       1          31
## Dickinson College                                       1          39
## Dickinson State University                              1          28
## Dillard University                                      1          12
## Doane College                                           1          42
## Dominican College of Blauvelt                           1           5
## Dordt College                                           1          17
## Dowling College                                         1           7
## Drake University                                        1          24
## Drew University                                         1          28
## Drury College                                           1          35
## Duke University                                         1          44
## Earlham College                                         1          46
## East Carolina University                                1          18
## East Tennessee State University                         1           9
## East Texas Baptist University                           1           7
## Eastern College                                         1          22
## Eastern Connecticut State University                    1          14
## Eastern Illinois University                             1           5
## Eastern Mennonite College                               1          29
## Eastern Nazarene College                                1          17
## Eckerd College                                          1          26
## Elizabethtown College                                   1          25
## Elmira College                                          1          21
## Elms College                                            1          21
## Elon College                                            1          34
## Embry Riddle Aeronautical University                    1           4
## Emory & Henry College                                   1          51
## Emory University                                        1          28
## Emporia State University                                1           4
## Erskine College                                         1          47
## Eureka College                                          1          31
## Evergreen State College                                 1          14
## Fairfield University                                    1          30
## Fayetteville State University                           1          10
## Ferrum College                                          1           9
## Flagler College                                         1           9
## Florida Institute of Technology                         1           7
## Florida International University                        1          20
## Florida Southern College                                1          10
## Florida State University                                1          15
## Fontbonne College                                       1          24
## Fordham University                                      1          14
## Fort Lewis College                                      1           6
## Francis Marion University                               1           8
## Franciscan University of Steubenville                   1           8
## Franklin College                                        1          37
## Franklin Pierce College                                 1          16
## Freed-Hardeman University                               1          13
## Fresno Pacific College                                  1          14
## Furman University                                       1          28
## Gannon University                                       1          18
## Gardner Webb University                                 1          12
## Geneva College                                          1          26
## George Fox College                                      1          22
## George Mason University                                 1           7
## George Washington University                            1          15
## Georgetown College                                      1          28
## Georgetown University                                   1          27
## Georgia Institute of Technology                         1          33
## Georgia State University                                1          10
## Georgian Court College                                  1          27
## Gettysburg College                                      1          32
## Goldey Beacom College                                   1           4
## Gonzaga University                                      1          32
## Gordon College                                          1          32
## Goshen College                                          1          46
## Goucher College                                         1          34
## Grace College and Seminary                              1          26
## Graceland College                                       1          24
## Grand Valley State University                           1           9
## Green Mountain College                                  1          24
## Greensboro College                                      1          31
## Greenville College                                      1          16
## Grinnell College                                        1          54
## Grove City College                                      1          18
## Guilford College                                        1          30
## Gustavus Adolphus College                               1          58
## Gwynedd Mercy College                                   1          22
## Hamilton College                                        1          60
## Hamline University                                      1          33
## Hampden - Sydney College                                1          53
## Hampton University                                      1           9
## Hanover College                                         1          26
## Hardin-Simmons University                               1          10
## Harding University                                      1          37
## Hartwick College                                        1          32
## Harvard University                                      1          52
## Harvey Mudd College                                     1          46
## Hastings College                                        1          17
## Hendrix College                                         1          26
## Hillsdale College                                       1          31
## Hiram College                                           1          34
## Hobart and William Smith Colleges                       1          37
## Hofstra University                                      1          10
## Hollins College                                         1          48
## Hood College                                            1          34
## Hope College                                            1          40
## Houghton College                                        1          24
## Huntingdon College                                      1           9
## Huntington College                                      1          25
## Huron University                                        1           4
## Husson College                                          1           4
## Illinois Benedictine College                            1          29
## Illinois College                                        1          30
## Illinois Institute of Technology                        1          26
## Illinois State University                               1          16
## Illinois Wesleyan University                            1          34
## Immaculata College                                      1          33
## Incarnate Word College                                  1          21
## Indiana State University                                1           8
## Indiana University at Bloomington                       1          24
## Indiana Wesleyan University                             1          15
## Iona College                                            1          14
## Iowa State University                                   1          22
## Ithaca College                                          1          25
## James Madison University                                1          29
## Jamestown College                                       1          21
## Jersey City State College                               1          10
## John Brown University                                   1          19
## John Carroll University                                 1          28
## Johns Hopkins University                                1          38
## Johnson State College                                   1          15
## Judson College                                          1          30
## Juniata College                                         1          37
## Kansas State University                                 1          22
## Kansas Wesleyan University                              1          14
## Keene State College                                     1          13
## Kentucky Wesleyan College                               1          32
## Kenyon College                                          1          46
## Keuka College                                           1          43
## King's College                                          1          37
## King College                                            1          25
## Knox College                                            1          33
## La Roche College                                        1          14
## La Salle University                                     1           9
## Lafayette College                                       1          38
## LaGrange College                                        1          12
## Lake Forest College                                     1          19
## Lakeland College                                        1          25
## Lamar University                                        1          12
## Lambuth University                                      1          10
## Lander University                                       1          11
## Lawrence University                                     1          57
## Le Moyne College                                        1          28
## Lebanon Valley College                                  1          30
## Lehigh University                                       1          43
## Lenoir-Rhyne College                                    1          20
## Lesley College                                          1          18
## LeTourneau University                                   1          23
## Lewis and Clark College                                 1          21
## Lewis University                                        1          10
## Lincoln Memorial University                             1          35
## Lincoln University                                      1           8
## Lindenwood College                                      1           9
## Linfield College                                        1          34
## Livingstone College                                     1          16
## Lock Haven University of Pennsylvania                   1          14
## Longwood College                                        1          23
## Loras College                                           1          24
## Louisiana College                                       1          11
## Louisiana State University at Baton Rouge               1          11
## Louisiana Tech University                               1          13
## Loyola College                                          1          27
## Loyola Marymount University                             1          10
## Loyola University                                       1          14
## Loyola University Chicago                               1          15
## Luther College                                          1          38
## Lycoming College                                        1          32
## Lynchburg College                                       1          24
## Lyndon State College                                    1          15
## Macalester College                                      1          37
## MacMurray College                                       1          33
## Malone College                                          1          16
## Manchester College                                      1          20
## Manhattan College                                       1          25
## Manhattanville College                                  1          24
## Mankato State University                                1          11
## Marian College of Fond du Lac                           1          21
## Marietta College                                        1          30
## Marist College                                          1          34
## Marquette University                                    1          25
## Marshall University                                     1          10
## Mary Baldwin College                                    1          50
## Mary Washington College                                 1          30
## Marymount College Tarrytown                             1          30
## Marymount Manhattan College                             1          20
## Marymount University                                    1          17
## Maryville College                                       1          43
## Maryville University                                    1          13
## Marywood College                                        1          30
## Massachusetts Institute of Technology                   1          35
## Mayville State University                               1          11
## McKendree College                                       1          21
## McMurry University                                      1          11
## McPherson College                                       1          45
## Mercer University                                       1          15
## Mercyhurst College                                      1          29
## Meredith College                                        1          33
## Merrimack College                                       1          22
## Mesa State College                                      1          12
## Messiah College                                         1          30
## Miami University at Oxford                              1          20
## Michigan State University                               1           9
## Michigan Technological University                       1          25
## MidAmerica Nazarene College                             1          20
## Millersville University of Penn.                        1          20
## Milligan College                                        1          16
## Millikin University                                     1          25
## Millsaps College                                        1          38
## Milwaukee School of Engineering                         1          23
## Mississippi College                                     1          18
## Mississippi State University                            1          20
## Mississippi University for Women                        1           8
## Missouri Southern State College                         1           9
## Missouri Valley College                                 1          16
## Monmouth College IL                                     1          43
## Monmouth College                                        1          15
## Montana College of Mineral Sci. & Tech.                 1          31
## Montana State University                                1           8
## Montclair State University                              1           9
## Montreat-Anderson College                               1           4
## Moorhead State University                               1          19
## Moravian College                                        1          28
## Morehouse College                                       1          10
## Morningside College                                     1          32
## Morris College                                          1          34
## Mount Holyoke College                                   1          51
## Mount Marty College                                     1          38
## Mount Mary College                                      1          26
## Mount Mercy College                                     1          30
## Mount Saint Clare College                               1          43
## Mount Saint Mary's College                              1          36
## Mount Saint Mary College                                1          23
## Mount St. Mary's College                                1          19
## Mount Union College                                     1          35
## Mount Vernon Nazarene College                           1           7
## Muhlenberg College                                      1          39
## Murray State University                                 1          27
## Muskingum College                                       1          27
## National-Louis University                               1           2
## Nazareth College of Rochester                           1          24
## New Jersey Institute of Technology                      1          19
## New Mexico Institute of Mining and Tech.                1          11
## New York University                                     1          16
## Newberry College                                        1          32
## Niagara University                                      1          20
## North Adams State College                               1          17
## North Carolina A. & T. State University                 1           9
## North Carolina State University at Raleigh              1          21
## North Carolina Wesleyan College                         1          11
## North Central College                                   1          33
## North Dakota State University                           1          24
## North Park College                                      1          24
## Northeast Missouri State University                     1          13
## Northeastern University                                 1          17
## Northern Arizona University                             1           7
## Northern Illinois University                            1          11
## Northwest Missouri State University                     1          23
## Northwest Nazarene College                              1          20
## Northwestern College                                    1          34
## Northwestern University                                 1          25
## Norwich University                                      1          22
## Notre Dame College                                      1          26
## Oakland University                                      1          13
## Oberlin College                                         1          47
## Occidental College                                      1          30
## Oglethorpe University                                   1          27
## Ohio Northern University                                1          31
## Ohio University                                         1          13
## Ohio Wesleyan University                                1          32
## Oklahoma Baptist University                             1          18
## Oklahoma Christian University                           1           8
## Oklahoma State University                               1          14
## Otterbein College                                       1          30
## Ouachita Baptist University                             1          10
## Our Lady of the Lake University                         1          25
## Pace University                                         1          10
## Pacific Lutheran University                             1          23
## Pacific Union College                                   1          12
## Pacific University                                      1          22
## Pembroke State University                               1           5
## Pennsylvania State Univ. Main Campus                    1          19
## Pepperdine University                                   1          13
## Peru State College                                      1          24
## Pfeiffer College                                        1          13
## Philadelphia Coll. of Textiles and Sci.                 1          15
## Phillips University                                     1          19
## Piedmont College                                        1          17
## Pikeville College                                       1          14
## Pitzer College                                          1          11
## Point Loma Nazarene College                             1          19
## Point Park College                                      1          10
## Polytechnic University                                  1          14
## Prairie View A. and M. University                       1           1
## Presbyterian College                                    1          42
## Princeton University                                    1          54
## Providence College                                      1          35
## Purdue University at West Lafayette                     1          15
## Queens College                                          1          36
## Quincy University                                       1          32
## Quinnipiac College                                      1          33
## Radford University                                      1           9
## Ramapo College of New Jersey                            1           8
## Randolph-Macon College                                  1          38
## Randolph-Macon Woman's College                          1          24
## Reed College                                            1          37
## Regis College                                           1          37
## Rensselaer Polytechnic Institute                        1          21
## Rhodes College                                          1          47
## Rider University                                        1          23
## Ripon College                                           1          49
## Rivier College                                          1          19
## Roanoke College                                         1          26
## Rockhurst College                                       1          21
## Rocky Mountain College                                  1          27
## Roger Williams University                               1           8
## Rollins College                                         1          23
## Rosary College                                          1          30
## Rowan College of New Jersey                             1           6
## Rutgers at New Brunswick                                1          19
## Rutgers State University at Camden                      1          12
## Rutgers State University at Newark                      1          16
## Sacred Heart University                                 1          16
## Saint Ambrose University                                1          16
## Saint Anselm College                                    1          29
## Saint Cloud State University                            1          10
## Saint Francis College IN                                1           4
## Saint Francis College                                   1          24
## Saint John's University                                 1          38
## Saint Joseph's College IN                               1          19
## Saint Joseph's College                                  1           8
## Saint Joseph's University                               1          13
## Saint Joseph College                                    1          32
## Saint Louis University                                  1          19
## Saint Mary's College                                    1          31
## Saint Mary's College of Minnesota                       1          19
## Saint Mary-of-the-Woods College                         1          37
## Saint Michael's College                                 1          34
## Saint Olaf College                                      1          31
## Saint Peter's College                                   1          22
## Saint Vincent College                                   1          31
## Saint Xavier University                                 1          15
## Salem-Teikyo University                                 1           9
## Salem College                                           1          46
## Salisbury State University                              1          18
## Samford University                                      1          17
## San Diego State University                              1           7
## Santa Clara University                                  1          19
## Sarah Lawrence College                                  1          18
## Savannah Coll. of Art and Design                        1          26
## Schreiner College                                       1          23
## Scripps College                                         1          41
## Seattle Pacific University                              1          20
## Seattle University                                      1          16
## Seton Hall University                                   1          15
## Seton Hill College                                      1          37
## Shippensburg University of Penn.                        1          13
## Shorter College                                         1          18
## Siena College                                           1          42
## Siena Heights College                                   1          17
## Simmons College                                         1          33
## Simpson College                                         1          36
## Sioux Falls College                                     1           7
## Skidmore College                                        1          29
## Smith College                                           1          44
## South Dakota State University                           1          29
## Southeast Missouri State University                     1           8
## Southeastern Oklahoma State Univ.                       1           9
## Southern California College                             1          11
## Southern Illinois University at Edwardsville            1           8
## Southern Methodist University                           1          17
## Southwest Baptist University                            1          13
## Southwest Missouri State University                     1          11
## Southwest State University                              1          31
## Southwestern Adventist College                          1          12
## Southwestern College                                    1          12
## Southwestern University                                 1          35
## Spalding University                                     1          40
## Spelman College                                         1          18
## Spring Arbor College                                    1           9
## St. Bonaventure University                              1          32
## St. John's College                                      1          26
## St. John Fisher College                                 1          29
## St. Lawrence University                                 1          38
## St. Martin's College                                    1           8
## St. Mary's College of California                        1          17
## St. Mary's College of Maryland                          1          23
## St. Mary's University of San Antonio                    1           7
## St. Norbert College                                     1          36
## St. Paul's College                                      1           8
## St. Thomas Aquinas College                              1          13
## Stephens College                                        1          17
## Stetson University                                      1          24
## Stevens Institute of Technology                         1          33
## Stockton College of New Jersey                          1           7
## Stonehill College                                       1          30
## SUNY at Albany                                          1          16
## SUNY at Binghamton                                      1          15
## SUNY at Buffalo                                         1          15
## SUNY at Stony Brook                                     1           7
## SUNY College  at Brockport                              1          14
## SUNY College  at Oswego                                 1          21
## SUNY College at Buffalo                                 1          12
## SUNY College at Cortland                                1          17
## SUNY College at Fredonia                                1          10
## SUNY College at Geneseo                                 1          25
## SUNY College at New Paltz                               1           8
## SUNY College at Plattsburgh                             1          16
## SUNY College at Potsdam                                 1          17
## SUNY College at Purchase                                1           8
## Susquehanna University                                  1          37
## Sweet Briar College                                     1          48
## Syracuse University                                     1          13
## Tabor College                                           1          15
## Talladega College                                       1           7
## Taylor University                                       1          32
## Tennessee Wesleyan College                              1          16
## Texas A&M Univ. at College Station                      1          29
## Texas A&M University at Galveston                       1          16
## Texas Christian University                              1          23
## Texas Lutheran College                                  1          24
## Texas Southern University                               1          21
## Texas Wesleyan University                               1          10
## The Citadel                                             1          17
## Thiel College                                           1          16
## Tiffin University                                       1          40
## Transylvania University                                 1          41
## Trenton State College                                   1           6
## Tri-State University                                    1          24
## Trinity College CT                                      1          48
## Trinity College DC                                      1          37
## Trinity College VT                                      1          26
## Trinity University                                      1          20
## Tulane University                                       1          21
## Tusculum College                                        1          28
## Tuskegee University                                     1           7
## Union College KY                                        1           9
## Union College NY                                        1          49
## Univ. of Wisconsin at OshKosh                           1          14
## University of Alabama at Birmingham                     1          16
## University of Arkansas at Fayetteville                  1          10
## University of California at Berkeley                    1          10
## University of California at Irvine                      1          11
## University of Central Florida                           1           9
## University of Charleston                                1          10
## University of Chicago                                   1          36
## University of Cincinnati                                1           6
## University of Connecticut at Storrs                     1          16
## University of Dallas                                    1          26
## University of Dayton                                    1          25
## University of Delaware                                  1          15
## University of Denver                                    1          21
## University of Detroit Mercy                             1          14
## University of Dubuque                                   1          18
## University of Evansville                                1          26
## University of Florida                                   1          20
## University of Georgia                                   1          22
## University of Hartford                                  1           9
## University of Hawaii at Manoa                           1           6
## University of Illinois - Urbana                         1          13
## University of Illinois at Chicago                       1           6
## University of Indianapolis                              1          23
## University of Kansas                                    1          17
## University of La Verne                                  1          23
## University of Louisville                                1          24
## University of Maine at Farmington                       1          26
## University of Maine at Machias                          1           4
## University of Maine at Presque Isle                     1          11
## University of Maryland at Baltimore County              1           6
## University of Maryland at College Park                  1          12
## University of Massachusetts at Amherst                  1          15
## University of Massachusetts at Dartmouth                1          20
## University of Miami                                     1          17
## University of Michigan at Ann Arbor                     1          26
## University of Minnesota at Duluth                       1          11
## University of Minnesota at Morris                       1          16
## University of Minnesota Twin Cities                     1          37
## University of Mississippi                               1          14
## University of Missouri at Columbia                      1          15
## University of Missouri at Rolla                         1          23
## University of Missouri at Saint Louis                   1          15
## University of Mobile                                    1           4
## University of Montevallo                                1           8
## University of Nebraska at Lincoln                       1          48
## University of New England                               1          13
## University of New Hampshire                             1          16
## University of North Carolina at Asheville               1          11
## University of North Carolina at Chapel Hill             1          23
## University of North Carolina at Charlotte               1           7
## University of North Carolina at Greensboro              1          17
## University of North Carolina at Wilmington              1          15
## University of North Dakota                              1          16
## University of North Florida                             1          14
## University of North Texas                               1           6
## University of Northern Colorado                         1           8
## University of Northern Iowa                             1          26
## University of Notre Dame                                1          46
## University of Oklahoma                                  1          11
## University of Oregon                                    1          13
## University of Pennsylvania                              1          38
## University of Pittsburgh-Main Campus                    1          10
## University of Portland                                  1          17
## University of Puget Sound                               1          17
## University of Rhode Island                              1           7
## University of Richmond                                  1          32
## University of Rochester                                 1          23
## University of San Diego                                 1          13
## University of San Francisco                             1           8
## University of Sci. and Arts of Oklahoma                 1           3
## University of Scranton                                  1          41
## University of South Carolina at Aiken                   1           3
## University of South Carolina at Columbia                1          18
## University of South Florida                             1           7
## University of Southern California                       1          10
## University of Southern Colorado                         1           0
## University of Southern Indiana                          1          21
## University of Southern Mississippi                      1          23
## University of St. Thomas MN                             1          13
## University of St. Thomas TX                             1          17
## University of Tennessee at Knoxville                    1          22
## University of Texas at Arlington                        1           4
## University of Texas at Austin                           1          11
## University of Texas at San Antonio                      1           3
## University of the Arts                                  1           9
## University of the Pacific                               1          14
## University of the South                                 1          52
## University of Tulsa                                     1          10
## University of Utah                                      1           9
## University of Vermont                                   1          10
## University of Virginia                                  1          22
## University of Washington                                1          10
## University of West Florida                              1          12
## University of Wisconsin-Stout                           1          17
## University of Wisconsin-Superior                        1          15
## University of Wisconsin-Whitewater                      1          16
## University of Wisconsin at Green Bay                    1           1
## University of Wisconsin at Madison                      1          20
## University of Wisconsin at Milwaukee                    1           8
## University of Wyoming                                   1          13
## Upper Iowa University                                   1          19
## Ursinus College                                         1          40
## Ursuline College                                        1          15
## Valley City State University                            1          25
## Valparaiso University                                   1          23
## Vanderbilt University                                   1          26
## Vassar College                                          1          39
## Villanova University                                    1          24
## Virginia Commonwealth University                        1          11
## Virginia State University                               1          11
## Virginia Tech                                           1          20
## Virginia Union University                               1           8
## Virginia Wesleyan College                               1          14
## Viterbo College                                         1          31
## Voorhees College                                        1           3
## Wabash College                                          1          55
## Wagner College                                          1          23
## Wake Forest University                                  1          37
## Walsh University                                        1          33
## Warren Wilson College                                   1          20
## Wartburg College                                        1          37
## Washington and Jefferson College                        1          40
## Washington and Lee University                           1          45
## Washington College                                      1          37
## Washington State University                             1          30
## Washington University                                   1          31
## Wayne State College                                     1          29
## Waynesburg College                                      1          26
## Webber College                                          1           4
## Webster University                                      1          14
## Wellesley College                                       1          51
## Wells College                                           1          42
## Wentworth Institute of Technology                       1           8
## Wesley College                                          1          15
## Wesleyan University                                     1          39
## West Chester University of Penn.                        1          16
## West Liberty State College                              1          10
## West Virginia Wesleyan College                          1          42
## Western Carolina University                             1           9
## Western Maryland College                                1          39
## Western Michigan University                             1          11
## Western New England College                             1          15
## Western State College of Colorado                       1           4
## Western Washington University                           1          10
## Westfield State College                                 1          20
## Westminster College MO                                  1          20
## Westminster College                                     1          41
## Westminster College of Salt Lake City                   1          34
## Westmont College                                        1          17
## Wheaton College IL                                      1          40
## Westminster College PA                                  1          41
## Wheeling Jesuit College                                 1          27
## Whitman College                                         1          51
## Whittier College                                        1          29
## Whitworth College                                       1          20
## Widener University                                      1          19
## Wilkes University                                       1          24
## Willamette University                                   1          37
## William Jewell College                                  1          19
## William Woods University                                1          16
## Williams College                                        1          64
## Wilson College                                          1          43
## Wingate College                                         1           8
## Winona State University                                 1          18
## Winthrop University                                     1          26
## Wisconsin Lutheran College                              1          26
## Wittenberg University                                   1          29
## Wofford College                                         1          42
## Worcester Polytechnic Institute                         1          34
## Worcester State College                                 1          14
## Xavier University                                       1          31
## Xavier University of Louisiana                          1          20
## Yale University                                         1          49
## York College of Pennsylvania                            1          28
##                                               I(perc.alumni^2)
## Abilene Christian University                               144
## Adelphi University                                         256
## Adrian College                                             900
## Agnes Scott College                                       1369
## Alaska Pacific University                                    4
## Albertson College                                          121
## Albertus Magnus College                                    676
## Albion College                                            1369
## Albright College                                           529
## Alderson-Broaddus College                                  225
## Alfred University                                          961
## Allegheny College                                         1681
## Allentown Coll. of St. Francis de Sales                    441
## Alma College                                              1024
## Alverno College                                            676
## American International College                             361
## Amherst College                                           3969
## Anderson University                                        196
## Andrews University                                         324
## Angelo State University                                     25
## Antioch University                                        1225
## Appalachian State University                               196
## Aquinas College                                            625
## Arizona State University Main campus                        25
## Arkansas College (Lyon College)                            576
## Arkansas Tech University                                    25
## Assumption College                                         900
## Auburn University-Main Campus                              324
## Augsburg College                                           961
## Augustana College IL                                      1600
## Augustana College                                          900
## Austin College                                            1089
## Averett College                                            121
## Baker University                                           441
## Baldwin-Wallace College                                    400
## Barat College                                             1225
## Bard College                                               900
## Barnard College                                           1089
## Barry University                                           121
## Baylor University                                         1444
## Beaver College                                             900
## Bellarmine College                                         961
## Belmont Abbey College                                      100
## Belmont University                                         361
## Beloit College                                             676
## Bemidji State University                                   256
## Benedictine College                                        324
## Bennington College                                        1089
## Bentley College                                            400
## Berry College                                              289
## Bethany College                                            841
## Bethel College KS                                         1024
## Bethel College                                             169
## Bethune Cookman College                                     81
## Birmingham-Southern College                               1156
## Blackburn College                                         2809
## Bloomsburg Univ. of Pennsylvania                           361
## Bluefield College                                            9
## Bluffton College                                           361
## Boston University                                          256
## Bowdoin College                                           2704
## Bowling Green State University                             196
## Bradford College                                           441
## Bradley University                                         441
## Brandeis University                                        576
## Brenau University                                          144
## Brewton-Parker College                                     100
## Briar Cliff College                                        676
## Bridgewater College                                        576
## Brigham Young University at Provo                         1600
## Brown University                                          1521
## Bryn Mawr College                                         2401
## Bucknell University                                       1296
## Buena Vista College                                        100
## Butler University                                          841
## Cabrini College                                           1296
## Caldwell College                                           625
## California Lutheran University                             289
## California Polytechnic-San Luis                            169
## California State University at Fresno                       64
## Calvin College                                            1681
## Campbell University                                       1156
## Campbellsville College                                     169
## Canisius College                                           676
## Capital University                                         729
## Capitol College                                            576
## Carleton College                                          3600
## Carnegie Mellon University                                 961
## Carroll College                                            625
## Carson-Newman College                                      256
## Carthage College                                           484
## Case Western Reserve University                            841
## Castleton State College                                     64
## Catawba College                                            729
## Catholic University of America                             324
## Cazenovia College                                          400
## Cedar Crest College                                       1521
## Cedarville College                                        1156
## Centenary College                                          400
## Centenary College of Louisiana                             625
## Center for Creative Studies                                 16
## Central College                                            841
## Central Connecticut State University                        16
## Central Missouri State University                           16
## Central Washington University                                0
## Central Wesleyan College                                   324
## Centre College                                            3600
## Chapman University                                          36
## Chatham College                                           1369
## Chestnut Hill College                                      841
## Christendom College                                        289
## Christian Brothers University                              576
## Christopher Newport University                             256
## Claflin College                                            961
## Claremont McKenna College                                 2704
## Clark University                                          1225
## Clarke College                                             729
## Clarkson University                                       1024
## Clemson University                                         289
## Clinch Valley Coll. of  the Univ. of Virginia               81
## Coe College                                               1024
## Coker College                                             1521
## Colby College                                             1681
## Colgate University                                        2025
## College Misericordia                                       529
## College of Charleston                                      324
## College of Mount St. Joseph                               1225
## College of Mount St. Vincent                              1225
## College of Notre Dame                                       49
## College of Notre Dame of Maryland                         1024
## College of Saint Benedict                                  676
## College of Saint Catherine                                1024
## College of Saint Elizabeth                                 529
## College of Saint Rose                                      784
## College of Santa Fe                                         49
## College of St. Joseph                                      361
## College of St. Scholastica                                1089
## College of the Holy Cross                                 3025
## College of William and Mary                                961
## College of Wooster                                        1849
## Colorado College                                          2601
## Colorado State University                                  100
## Columbia College MO                                          4
## Columbia College                                          1156
## Columbia University                                        441
## Concordia College at St. Paul                              324
## Concordia Lutheran College                                  81
## Concordia University CA                                    169
## Concordia University                                       169
## Connecticut College                                       1600
## Converse College                                           961
## Cornell College                                            961
## Creighton University                                      1024
## Culver-Stockton College                                    784
## Cumberland College                                          16
## D'Youville College                                        1764
## Dana College                                               625
## Daniel Webster College                                     100
## Dartmouth College                                         2401
## Davidson College                                          2116
## Defiance College                                           361
## Delta State University                                     256
## Denison University                                        2025
## DePauw University                                          961
## Dickinson College                                         1521
## Dickinson State University                                 784
## Dillard University                                         144
## Doane College                                             1764
## Dominican College of Blauvelt                               25
## Dordt College                                              289
## Dowling College                                             49
## Drake University                                           576
## Drew University                                            784
## Drury College                                             1225
## Duke University                                           1936
## Earlham College                                           2116
## East Carolina University                                   324
## East Tennessee State University                             81
## East Texas Baptist University                               49
## Eastern College                                            484
## Eastern Connecticut State University                       196
## Eastern Illinois University                                 25
## Eastern Mennonite College                                  841
## Eastern Nazarene College                                   289
## Eckerd College                                             676
## Elizabethtown College                                      625
## Elmira College                                             441
## Elms College                                               441
## Elon College                                              1156
## Embry Riddle Aeronautical University                        16
## Emory & Henry College                                     2601
## Emory University                                           784
## Emporia State University                                    16
## Erskine College                                           2209
## Eureka College                                             961
## Evergreen State College                                    196
## Fairfield University                                       900
## Fayetteville State University                              100
## Ferrum College                                              81
## Flagler College                                             81
## Florida Institute of Technology                             49
## Florida International University                           400
## Florida Southern College                                   100
## Florida State University                                   225
## Fontbonne College                                          576
## Fordham University                                         196
## Fort Lewis College                                          36
## Francis Marion University                                   64
## Franciscan University of Steubenville                       64
## Franklin College                                          1369
## Franklin Pierce College                                    256
## Freed-Hardeman University                                  169
## Fresno Pacific College                                     196
## Furman University                                          784
## Gannon University                                          324
## Gardner Webb University                                    144
## Geneva College                                             676
## George Fox College                                         484
## George Mason University                                     49
## George Washington University                               225
## Georgetown College                                         784
## Georgetown University                                      729
## Georgia Institute of Technology                           1089
## Georgia State University                                   100
## Georgian Court College                                     729
## Gettysburg College                                        1024
## Goldey Beacom College                                       16
## Gonzaga University                                        1024
## Gordon College                                            1024
## Goshen College                                            2116
## Goucher College                                           1156
## Grace College and Seminary                                 676
## Graceland College                                          576
## Grand Valley State University                               81
## Green Mountain College                                     576
## Greensboro College                                         961
## Greenville College                                         256
## Grinnell College                                          2916
## Grove City College                                         324
## Guilford College                                           900
## Gustavus Adolphus College                                 3364
## Gwynedd Mercy College                                      484
## Hamilton College                                          3600
## Hamline University                                        1089
## Hampden - Sydney College                                  2809
## Hampton University                                          81
## Hanover College                                            676
## Hardin-Simmons University                                  100
## Harding University                                        1369
## Hartwick College                                          1024
## Harvard University                                        2704
## Harvey Mudd College                                       2116
## Hastings College                                           289
## Hendrix College                                            676
## Hillsdale College                                          961
## Hiram College                                             1156
## Hobart and William Smith Colleges                         1369
## Hofstra University                                         100
## Hollins College                                           2304
## Hood College                                              1156
## Hope College                                              1600
## Houghton College                                           576
## Huntingdon College                                          81
## Huntington College                                         625
## Huron University                                            16
## Husson College                                              16
## Illinois Benedictine College                               841
## Illinois College                                           900
## Illinois Institute of Technology                           676
## Illinois State University                                  256
## Illinois Wesleyan University                              1156
## Immaculata College                                        1089
## Incarnate Word College                                     441
## Indiana State University                                    64
## Indiana University at Bloomington                          576
## Indiana Wesleyan University                                225
## Iona College                                               196
## Iowa State University                                      484
## Ithaca College                                             625
## James Madison University                                   841
## Jamestown College                                          441
## Jersey City State College                                  100
## John Brown University                                      361
## John Carroll University                                    784
## Johns Hopkins University                                  1444
## Johnson State College                                      225
## Judson College                                             900
## Juniata College                                           1369
## Kansas State University                                    484
## Kansas Wesleyan University                                 196
## Keene State College                                        169
## Kentucky Wesleyan College                                 1024
## Kenyon College                                            2116
## Keuka College                                             1849
## King's College                                            1369
## King College                                               625
## Knox College                                              1089
## La Roche College                                           196
## La Salle University                                         81
## Lafayette College                                         1444
## LaGrange College                                           144
## Lake Forest College                                        361
## Lakeland College                                           625
## Lamar University                                           144
## Lambuth University                                         100
## Lander University                                          121
## Lawrence University                                       3249
## Le Moyne College                                           784
## Lebanon Valley College                                     900
## Lehigh University                                         1849
## Lenoir-Rhyne College                                       400
## Lesley College                                             324
## LeTourneau University                                      529
## Lewis and Clark College                                    441
## Lewis University                                           100
## Lincoln Memorial University                               1225
## Lincoln University                                          64
## Lindenwood College                                          81
## Linfield College                                          1156
## Livingstone College                                        256
## Lock Haven University of Pennsylvania                      196
## Longwood College                                           529
## Loras College                                              576
## Louisiana College                                          121
## Louisiana State University at Baton Rouge                  121
## Louisiana Tech University                                  169
## Loyola College                                             729
## Loyola Marymount University                                100
## Loyola University                                          196
## Loyola University Chicago                                  225
## Luther College                                            1444
## Lycoming College                                          1024
## Lynchburg College                                          576
## Lyndon State College                                       225
## Macalester College                                        1369
## MacMurray College                                         1089
## Malone College                                             256
## Manchester College                                         400
## Manhattan College                                          625
## Manhattanville College                                     576
## Mankato State University                                   121
## Marian College of Fond du Lac                              441
## Marietta College                                           900
## Marist College                                            1156
## Marquette University                                       625
## Marshall University                                        100
## Mary Baldwin College                                      2500
## Mary Washington College                                    900
## Marymount College Tarrytown                                900
## Marymount Manhattan College                                400
## Marymount University                                       289
## Maryville College                                         1849
## Maryville University                                       169
## Marywood College                                           900
## Massachusetts Institute of Technology                     1225
## Mayville State University                                  121
## McKendree College                                          441
## McMurry University                                         121
## McPherson College                                         2025
## Mercer University                                          225
## Mercyhurst College                                         841
## Meredith College                                          1089
## Merrimack College                                          484
## Mesa State College                                         144
## Messiah College                                            900
## Miami University at Oxford                                 400
## Michigan State University                                   81
## Michigan Technological University                          625
## MidAmerica Nazarene College                                400
## Millersville University of Penn.                           400
## Milligan College                                           256
## Millikin University                                        625
## Millsaps College                                          1444
## Milwaukee School of Engineering                            529
## Mississippi College                                        324
## Mississippi State University                               400
## Mississippi University for Women                            64
## Missouri Southern State College                             81
## Missouri Valley College                                    256
## Monmouth College IL                                       1849
## Monmouth College                                           225
## Montana College of Mineral Sci. & Tech.                    961
## Montana State University                                    64
## Montclair State University                                  81
## Montreat-Anderson College                                   16
## Moorhead State University                                  361
## Moravian College                                           784
## Morehouse College                                          100
## Morningside College                                       1024
## Morris College                                            1156
## Mount Holyoke College                                     2601
## Mount Marty College                                       1444
## Mount Mary College                                         676
## Mount Mercy College                                        900
## Mount Saint Clare College                                 1849
## Mount Saint Mary's College                                1296
## Mount Saint Mary College                                   529
## Mount St. Mary's College                                   361
## Mount Union College                                       1225
## Mount Vernon Nazarene College                               49
## Muhlenberg College                                        1521
## Murray State University                                    729
## Muskingum College                                          729
## National-Louis University                                    4
## Nazareth College of Rochester                              576
## New Jersey Institute of Technology                         361
## New Mexico Institute of Mining and Tech.                   121
## New York University                                        256
## Newberry College                                          1024
## Niagara University                                         400
## North Adams State College                                  289
## North Carolina A. & T. State University                     81
## North Carolina State University at Raleigh                 441
## North Carolina Wesleyan College                            121
## North Central College                                     1089
## North Dakota State University                              576
## North Park College                                         576
## Northeast Missouri State University                        169
## Northeastern University                                    289
## Northern Arizona University                                 49
## Northern Illinois University                               121
## Northwest Missouri State University                        529
## Northwest Nazarene College                                 400
## Northwestern College                                      1156
## Northwestern University                                    625
## Norwich University                                         484
## Notre Dame College                                         676
## Oakland University                                         169
## Oberlin College                                           2209
## Occidental College                                         900
## Oglethorpe University                                      729
## Ohio Northern University                                   961
## Ohio University                                            169
## Ohio Wesleyan University                                  1024
## Oklahoma Baptist University                                324
## Oklahoma Christian University                               64
## Oklahoma State University                                  196
## Otterbein College                                          900
## Ouachita Baptist University                                100
## Our Lady of the Lake University                            625
## Pace University                                            100
## Pacific Lutheran University                                529
## Pacific Union College                                      144
## Pacific University                                         484
## Pembroke State University                                   25
## Pennsylvania State Univ. Main Campus                       361
## Pepperdine University                                      169
## Peru State College                                         576
## Pfeiffer College                                           169
## Philadelphia Coll. of Textiles and Sci.                    225
## Phillips University                                        361
## Piedmont College                                           289
## Pikeville College                                          196
## Pitzer College                                             121
## Point Loma Nazarene College                                361
## Point Park College                                         100
## Polytechnic University                                     196
## Prairie View A. and M. University                            1
## Presbyterian College                                      1764
## Princeton University                                      2916
## Providence College                                        1225
## Purdue University at West Lafayette                        225
## Queens College                                            1296
## Quincy University                                         1024
## Quinnipiac College                                        1089
## Radford University                                          81
## Ramapo College of New Jersey                                64
## Randolph-Macon College                                    1444
## Randolph-Macon Woman's College                             576
## Reed College                                              1369
## Regis College                                             1369
## Rensselaer Polytechnic Institute                           441
## Rhodes College                                            2209
## Rider University                                           529
## Ripon College                                             2401
## Rivier College                                             361
## Roanoke College                                            676
## Rockhurst College                                          441
## Rocky Mountain College                                     729
## Roger Williams University                                   64
## Rollins College                                            529
## Rosary College                                             900
## Rowan College of New Jersey                                 36
## Rutgers at New Brunswick                                   361
## Rutgers State University at Camden                         144
## Rutgers State University at Newark                         256
## Sacred Heart University                                    256
## Saint Ambrose University                                   256
## Saint Anselm College                                       841
## Saint Cloud State University                               100
## Saint Francis College IN                                    16
## Saint Francis College                                      576
## Saint John's University                                   1444
## Saint Joseph's College IN                                  361
## Saint Joseph's College                                      64
## Saint Joseph's University                                  169
## Saint Joseph College                                      1024
## Saint Louis University                                     361
## Saint Mary's College                                       961
## Saint Mary's College of Minnesota                          361
## Saint Mary-of-the-Woods College                           1369
## Saint Michael's College                                   1156
## Saint Olaf College                                         961
## Saint Peter's College                                      484
## Saint Vincent College                                      961
## Saint Xavier University                                    225
## Salem-Teikyo University                                     81
## Salem College                                             2116
## Salisbury State University                                 324
## Samford University                                         289
## San Diego State University                                  49
## Santa Clara University                                     361
## Sarah Lawrence College                                     324
## Savannah Coll. of Art and Design                           676
## Schreiner College                                          529
## Scripps College                                           1681
## Seattle Pacific University                                 400
## Seattle University                                         256
## Seton Hall University                                      225
## Seton Hill College                                        1369
## Shippensburg University of Penn.                           169
## Shorter College                                            324
## Siena College                                             1764
## Siena Heights College                                      289
## Simmons College                                           1089
## Simpson College                                           1296
## Sioux Falls College                                         49
## Skidmore College                                           841
## Smith College                                             1936
## South Dakota State University                              841
## Southeast Missouri State University                         64
## Southeastern Oklahoma State Univ.                           81
## Southern California College                                121
## Southern Illinois University at Edwardsville                64
## Southern Methodist University                              289
## Southwest Baptist University                               169
## Southwest Missouri State University                        121
## Southwest State University                                 961
## Southwestern Adventist College                             144
## Southwestern College                                       144
## Southwestern University                                   1225
## Spalding University                                       1600
## Spelman College                                            324
## Spring Arbor College                                        81
## St. Bonaventure University                                1024
## St. John's College                                         676
## St. John Fisher College                                    841
## St. Lawrence University                                   1444
## St. Martin's College                                        64
## St. Mary's College of California                           289
## St. Mary's College of Maryland                             529
## St. Mary's University of San Antonio                        49
## St. Norbert College                                       1296
## St. Paul's College                                          64
## St. Thomas Aquinas College                                 169
## Stephens College                                           289
## Stetson University                                         576
## Stevens Institute of Technology                           1089
## Stockton College of New Jersey                              49
## Stonehill College                                          900
## SUNY at Albany                                             256
## SUNY at Binghamton                                         225
## SUNY at Buffalo                                            225
## SUNY at Stony Brook                                         49
## SUNY College  at Brockport                                 196
## SUNY College  at Oswego                                    441
## SUNY College at Buffalo                                    144
## SUNY College at Cortland                                   289
## SUNY College at Fredonia                                   100
## SUNY College at Geneseo                                    625
## SUNY College at New Paltz                                   64
## SUNY College at Plattsburgh                                256
## SUNY College at Potsdam                                    289
## SUNY College at Purchase                                    64
## Susquehanna University                                    1369
## Sweet Briar College                                       2304
## Syracuse University                                        169
## Tabor College                                              225
## Talladega College                                           49
## Taylor University                                         1024
## Tennessee Wesleyan College                                 256
## Texas A&M Univ. at College Station                         841
## Texas A&M University at Galveston                          256
## Texas Christian University                                 529
## Texas Lutheran College                                     576
## Texas Southern University                                  441
## Texas Wesleyan University                                  100
## The Citadel                                                289
## Thiel College                                              256
## Tiffin University                                         1600
## Transylvania University                                   1681
## Trenton State College                                       36
## Tri-State University                                       576
## Trinity College CT                                        2304
## Trinity College DC                                        1369
## Trinity College VT                                         676
## Trinity University                                         400
## Tulane University                                          441
## Tusculum College                                           784
## Tuskegee University                                         49
## Union College KY                                            81
## Union College NY                                          2401
## Univ. of Wisconsin at OshKosh                              196
## University of Alabama at Birmingham                        256
## University of Arkansas at Fayetteville                     100
## University of California at Berkeley                       100
## University of California at Irvine                         121
## University of Central Florida                               81
## University of Charleston                                   100
## University of Chicago                                     1296
## University of Cincinnati                                    36
## University of Connecticut at Storrs                        256
## University of Dallas                                       676
## University of Dayton                                       625
## University of Delaware                                     225
## University of Denver                                       441
## University of Detroit Mercy                                196
## University of Dubuque                                      324
## University of Evansville                                   676
## University of Florida                                      400
## University of Georgia                                      484
## University of Hartford                                      81
## University of Hawaii at Manoa                               36
## University of Illinois - Urbana                            169
## University of Illinois at Chicago                           36
## University of Indianapolis                                 529
## University of Kansas                                       289
## University of La Verne                                     529
## University of Louisville                                   576
## University of Maine at Farmington                          676
## University of Maine at Machias                              16
## University of Maine at Presque Isle                        121
## University of Maryland at Baltimore County                  36
## University of Maryland at College Park                     144
## University of Massachusetts at Amherst                     225
## University of Massachusetts at Dartmouth                   400
## University of Miami                                        289
## University of Michigan at Ann Arbor                        676
## University of Minnesota at Duluth                          121
## University of Minnesota at Morris                          256
## University of Minnesota Twin Cities                       1369
## University of Mississippi                                  196
## University of Missouri at Columbia                         225
## University of Missouri at Rolla                            529
## University of Missouri at Saint Louis                      225
## University of Mobile                                        16
## University of Montevallo                                    64
## University of Nebraska at Lincoln                         2304
## University of New England                                  169
## University of New Hampshire                                256
## University of North Carolina at Asheville                  121
## University of North Carolina at Chapel Hill                529
## University of North Carolina at Charlotte                   49
## University of North Carolina at Greensboro                 289
## University of North Carolina at Wilmington                 225
## University of North Dakota                                 256
## University of North Florida                                196
## University of North Texas                                   36
## University of Northern Colorado                             64
## University of Northern Iowa                                676
## University of Notre Dame                                  2116
## University of Oklahoma                                     121
## University of Oregon                                       169
## University of Pennsylvania                                1444
## University of Pittsburgh-Main Campus                       100
## University of Portland                                     289
## University of Puget Sound                                  289
## University of Rhode Island                                  49
## University of Richmond                                    1024
## University of Rochester                                    529
## University of San Diego                                    169
## University of San Francisco                                 64
## University of Sci. and Arts of Oklahoma                      9
## University of Scranton                                    1681
## University of South Carolina at Aiken                        9
## University of South Carolina at Columbia                   324
## University of South Florida                                 49
## University of Southern California                          100
## University of Southern Colorado                              0
## University of Southern Indiana                             441
## University of Southern Mississippi                         529
## University of St. Thomas MN                                169
## University of St. Thomas TX                                289
## University of Tennessee at Knoxville                       484
## University of Texas at Arlington                            16
## University of Texas at Austin                              121
## University of Texas at San Antonio                           9
## University of the Arts                                      81
## University of the Pacific                                  196
## University of the South                                   2704
## University of Tulsa                                        100
## University of Utah                                          81
## University of Vermont                                      100
## University of Virginia                                     484
## University of Washington                                   100
## University of West Florida                                 144
## University of Wisconsin-Stout                              289
## University of Wisconsin-Superior                           225
## University of Wisconsin-Whitewater                         256
## University of Wisconsin at Green Bay                         1
## University of Wisconsin at Madison                         400
## University of Wisconsin at Milwaukee                        64
## University of Wyoming                                      169
## Upper Iowa University                                      361
## Ursinus College                                           1600
## Ursuline College                                           225
## Valley City State University                               625
## Valparaiso University                                      529
## Vanderbilt University                                      676
## Vassar College                                            1521
## Villanova University                                       576
## Virginia Commonwealth University                           121
## Virginia State University                                  121
## Virginia Tech                                              400
## Virginia Union University                                   64
## Virginia Wesleyan College                                  196
## Viterbo College                                            961
## Voorhees College                                             9
## Wabash College                                            3025
## Wagner College                                             529
## Wake Forest University                                    1369
## Walsh University                                          1089
## Warren Wilson College                                      400
## Wartburg College                                          1369
## Washington and Jefferson College                          1600
## Washington and Lee University                             2025
## Washington College                                        1369
## Washington State University                                900
## Washington University                                      961
## Wayne State College                                        841
## Waynesburg College                                         676
## Webber College                                              16
## Webster University                                         196
## Wellesley College                                         2601
## Wells College                                             1764
## Wentworth Institute of Technology                           64
## Wesley College                                             225
## Wesleyan University                                       1521
## West Chester University of Penn.                           256
## West Liberty State College                                 100
## West Virginia Wesleyan College                            1764
## Western Carolina University                                 81
## Western Maryland College                                  1521
## Western Michigan University                                121
## Western New England College                                225
## Western State College of Colorado                           16
## Western Washington University                              100
## Westfield State College                                    400
## Westminster College MO                                     400
## Westminster College                                       1681
## Westminster College of Salt Lake City                     1156
## Westmont College                                           289
## Wheaton College IL                                        1600
## Westminster College PA                                    1681
## Wheeling Jesuit College                                    729
## Whitman College                                           2601
## Whittier College                                           841
## Whitworth College                                          400
## Widener University                                         361
## Wilkes University                                          576
## Willamette University                                     1369
## William Jewell College                                     361
## William Woods University                                   256
## Williams College                                          4096
## Wilson College                                            1849
## Wingate College                                             64
## Winona State University                                    324
## Winthrop University                                        676
## Wisconsin Lutheran College                                 676
## Wittenberg University                                      841
## Wofford College                                           1764
## Worcester Polytechnic Institute                           1156
## Worcester State College                                    196
## Xavier University                                          961
## Xavier University of Louisiana                             400
## Yale University                                           2401
## York College of Pennsylvania                               784
## attr(,"assign")
## [1] 0 1 2
vcov(ajuste2) ### Matriz de variâncias e covariâncias dos estimadores.
##                   (Intercept)   perc.alumni I(perc.alumni^2)
## (Intercept)       3.526090777 -0.2679031981     4.254628e-03
## perc.alumni      -0.267903198  0.0242157266    -4.217689e-04
## I(perc.alumni^2)  0.004254628 -0.0004217689     7.959571e-06
### Vamos visualizar o ajuste do modelo
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
    stat_smooth(method = "lm", formula = y ~ x + I(x^2)) +
    theme_bw(base_size = 14)

### Extraindo os intervalos de confiança (95%) para os parâmetros
confint(ajuste2)
##                       2.5 %       97.5 %
## (Intercept)      42.2100816 49.582406847
## perc.alumni       0.7804723  1.391423440
## I(perc.alumni^2) -0.0131900 -0.002113503
### vamos realizar algumas predições. Considere faculdades com os seguintes
### percentuais de ex-alunos contribuintes: 13, 28 e 45

new_data <- data.frame(perc.alumni = c(13,28,45))
predict(ajuste2, newdata = new_data, se.fit = TRUE)
## $fit
##        1        2        3 
## 58.72042 70.30381 79.26910 
## 
## $se.fit
##         1         2         3 
## 0.6820636 0.7449543 1.2061247 
## 
## $df
## [1] 774
## 
## $residual.scale
## [1] 14.91414
### Estimativas pontuais e erros padrões (nota: erros padrões para a resposta
### média)

predict(ajuste2, newdata = new_data, interval = 'confidence') 
##        fit      lwr      upr
## 1 58.72042 57.38151 60.05933
## 2 70.30381 68.84144 71.76618
## 3 79.26910 76.90143 81.63676
### Estimativas pontiais e intervalos de confiança (95%) para a resposta média.

predict(ajuste2, newdata = new_data, interval = 'prediction') 
##        fit      lwr       upr
## 1 58.72042 29.41286  88.02798
## 2 70.30381 40.99035  99.61727
## 3 79.26910 49.89656 108.64164
### Estimativas pontiais e intervalos de confiança (95%) para a predição
### de uma nova observação.
### Vamos plotar as bandas de confiança e predição. Para isso, vamos preparar 
### uma base de dados com os valores ajustados e os ICs(95%) para a resposta
### média e para as predições.

pred_int <- predict(ajuste2, interval="prediction")
med_int <- predict(ajuste2, interval="confidence")
data_pred <- data.frame(pred_lwr = pred_int[,'lwr'], pred_upr = pred_int[,'upr'],
                        med_lwr = med_int[,'lwr'], med_upr = med_int[,'upr'],
                        fit = med_int[,'fit'], perc.alumni = College$perc.alumni,
                        Grad.Rate = College$Grad.Rate)

ggplot(data_pred, aes(x = perc.alumni, y = Grad.Rate))+
    geom_point() +
    geom_line(aes(y=med_lwr), color = "red", linetype = "dashed", linewidth = 1.25) +
    geom_line(aes(y=med_upr), color = "red", linetype = "dashed", linewidth = 1.25) +
    geom_line(aes(y=pred_lwr), color = "green", linetype = "dashed", linewidth = 1.25) +
    geom_line(aes(y=pred_upr), color = "green", linetype = "dashed", linewidth = 1.25) +
    geom_line(aes(y=fit), color = "black", linetype = "dashed", linewidth = 1.25) +
    theme_bw(base_size = 14)
### Diagnóstico do ajuste (análise de resíduos)

### Vamos produzir alguns gráficos para os resíduos

fit_aj2 <- fitted(ajuste2) ### Vetor de valores ajustados
resid_aj2 <- rstandard(ajuste2) ### Vetor de resíduos padronizados

data_fit <- data.frame(y = College$Grad.Rate, fit_aj2, resid_aj2)

ggplot(data_fit, aes(x=y, y=fit_aj2)) + geom_point() + stat_smooth(method="lm") +
    theme_bw(base_size = 14)
### Gráfico de valores observados versus valores ajustados. 

ggplot(ajuste2, aes(x=fit_aj2, y=resid_aj2)) + geom_point() +
    stat_smooth(method="loess") + geom_hline(yintercept=0, col="red", linetype="dashed") +
    theme_bw(base_size = 14) +
    xlab('Valores ajustados') +
    ylab('Resíduos')
### Gráfico de resíduos versus valores ajustados

qqPlot(ajuste2)
### Gráfico quantil-quantil para os resíduos
### Parte 2 - Regressão linear múltipla, incluindo todas as variáveis da base como explicativas (exceto a taxa de formação, que é a resposta)

ajuste_p2 <- lm(Grad.Rate ~., data = College)
### Ajuste da regressão linear múltipla. A especificação "~." indica que
### todas as demais variáveis da base devem ser incluídas como explicativas.
### A título de ilustração, se quiséssemos ajustar um modelo apenas com as
### variáveis "Private", "Apps" e "Accept":
summary(ajuste_p2) 
## 
## Call:
## lm(formula = Grad.Rate ~ ., data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.897  -7.132  -0.292   7.213  54.056 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 33.8736716  4.8480858   6.987 6.15e-12 ***
## PrivateYes   3.3813758  1.6965147   1.993 0.046605 *  
## Apps         0.0012984  0.0004418   2.939 0.003390 ** 
## Accept      -0.0006961  0.0008627  -0.807 0.419995    
## Enroll       0.0021593  0.0023081   0.936 0.349814    
## Top10perc    0.0548964  0.0717587   0.765 0.444501    
## Top25perc    0.1351288  0.0549667   2.458 0.014179 *  
## F.Undergrad -0.0004712  0.0004008  -1.176 0.240138    
## P.Undergrad -0.0014836  0.0003902  -3.802 0.000155 ***
## Outstate     0.0010174  0.0002334   4.359 1.49e-05 ***
## Room.Board   0.0019143  0.0005908   3.240 0.001246 ** 
## Books       -0.0022205  0.0029168  -0.761 0.446739    
## Personal    -0.0016635  0.0007698  -2.161 0.031000 *  
## PhD          0.0872827  0.0568102   1.536 0.124859    
## Terminal    -0.0747023  0.0623172  -1.199 0.231002    
## S.F.Ratio    0.0758222  0.1593102   0.476 0.634254    
## perc.alumni  0.2793343  0.0491750   5.680 1.91e-08 ***
## Expend      -0.0004565  0.0001542  -2.961 0.003163 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.75 on 759 degrees of freedom
## Multiple R-squared:  0.4615, Adjusted R-squared:  0.4495 
## F-statistic: 38.27 on 17 and 759 DF,  p-value: < 2.2e-16
ajuste_p2 <- lm(Grad.Rate ~.-Top25perc, data = College)

## Todas menos a top 25. pq o top 10 e top 25 "se sobrepoe"
## "ajustado os efeitos das outras variaveis"

summary(ajuste_p2) 
## 
## Call:
## lm(formula = Grad.Rate ~ . - Top25perc, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -47.786  -7.120  -0.257   7.171  54.050 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 36.3495104  4.7580380   7.640 6.57e-14 ***
## PrivateYes   3.3824039  1.7021347   1.987 0.047264 *  
## Apps         0.0011831  0.0004407   2.685 0.007421 ** 
## Accept      -0.0003936  0.0008568  -0.459 0.646059    
## Enroll       0.0015802  0.0023036   0.686 0.492938    
## Top10perc    0.1951766  0.0436553   4.471 8.98e-06 ***
## F.Undergrad -0.0003934  0.0004009  -0.981 0.326837    
## P.Undergrad -0.0014830  0.0003915  -3.788 0.000164 ***
## Outstate     0.0010161  0.0002342   4.339 1.63e-05 ***
## Room.Board   0.0019288  0.0005927   3.254 0.001188 ** 
## Books       -0.0020595  0.0029258  -0.704 0.481692    
## Personal    -0.0016865  0.0007723  -2.184 0.029273 *  
## PhD          0.0876416  0.0569982   1.538 0.124558    
## Terminal    -0.0565580  0.0620836  -0.911 0.362585    
## S.F.Ratio    0.0725205  0.1598322   0.454 0.650153    
## perc.alumni  0.2896423  0.0491582   5.892 5.73e-09 ***
## Expend      -0.0005259  0.0001521  -3.459 0.000573 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.79 on 760 degrees of freedom
## Multiple R-squared:  0.4573, Adjusted R-squared:  0.4458 
## F-statistic: 40.02 on 16 and 760 DF,  p-value: < 2.2e-16

Exemplo de numero de comodos e area em preço do imovel. Em sepada os dois são significantes. No modelo tudo junto ajutado a area é positiva e os comodos são negativos (mais comodos numa mesma area, ou seja comodos menores, menos preço).

ajuste_p2 <- lm(Grad.Rate ~ Expend, data = College)

## Todas menos a top 25. pq o top 10 e top 25 "se sobrepoe"
## "ajustado os efeitos das outras variaveis"

summary(ajuste_p2) 
## 
## Call:
## lm(formula = Grad.Rate ~ Expend, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -60.179 -10.275   0.238  10.377  55.058 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 5.306e+01  1.194e+00   44.42   <2e-16 ***
## Expend      1.284e-03  1.088e-04   11.80   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.83 on 775 degrees of freedom
## Multiple R-squared:  0.1524, Adjusted R-squared:  0.1513 
## F-statistic: 139.3 on 1 and 775 DF,  p-value: < 2.2e-16
ajuste_p2_ilustrativo <- lm(Grad.Rate ~ Private + Apps + Accept, data = College)

summary(ajuste_p2_ilustrativo) 
## 
## Call:
## lm(formula = Grad.Rate ~ Private + Apps + Accept, data = College)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.513  -8.856   0.228  10.516  50.053 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 48.5068491  1.4308933  33.900  < 2e-16 ***
## PrivateYes  17.7697914  1.3831646  12.847  < 2e-16 ***
## Apps         0.0030589  0.0004228   7.235 1.12e-12 ***
## Accept      -0.0025493  0.0006842  -3.726 0.000209 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.09 on 773 degrees of freedom
## Multiple R-squared:  0.2316, Adjusted R-squared:  0.2287 
## F-statistic: 77.68 on 3 and 773 DF,  p-value: < 2.2e-16
### Resumo do ajuste. Observe as variáveis com efeito significativo e os respectivos sinais.
### Variáveis com p-valor (Pr(>|t|)) < 0.05 podem ser consideradas com efeito 
### significativo na taxa de formados. Desta forma, as variáveis com efeito
### significativo na taxa de formados são Private (Yes), Apps, Top25perc, Outstate,
### Room.Board, 0.2793343; Já as variáveis com efeito negativo na taxa de formados
### são P.Undergrad, Personal e Expend.
### Apenas para fins de discussão, vamos ajustar um modelo de regressão tendo como
### única variável explicativa o gasto por aluno.
ajuste_temp <- lm(Grad.Rate ~ Expend, data = College)
summary(ajuste_temp)

### Compare o efeito do gasto por aluno na taxa de formação produzida pelo
### modelo em que ajustamos também os efeitos das demais variáveis com o
### efeito produzido pelo modelo em que as demais variáveis não são consideradas.
### Qual a diferença? Como você a justifica?
par(mfrow = c(2,2))
plot(ajuste_p2, which = 1:4)
### Gráficos de resíduos.
### O gráfico do canto superior esquerdo indica que a variância dos resíduos varia
### um pouco com a média (fitted values), e que os resíduos apresentam algum desvio
### da normalidade (conforme o qqplot). Ainda, conforme o gráfico do canto inferior
### direito, algumas observações podem ser identificadas como possivelmente influentes,
### produzindo maiores valores para a distância de Cook.

### Embora tenhamos alguns indicativos (ainda que não tão severos) de falta de ajuste
### da regressão linear, para fins didáticos vamos seguir a análise com esse tipo
### de modelagem.
### Agora, vamos para a etapa de seleção de covariáveis. Para isso, vamos
### usar os recursos do pacote leaps. Vamos consultar a documentação da função
### regsubsets.

help("regsubsets")

all_reg <- regsubsets(Grad.Rate ~ ., method = "exhaustive", nvmax = 18, data = College)
### Explorando TODAS as regressões possíveis
all_reg
plot(all_reg, scale="r2") ### Resultados baseados no R2
plot(all_reg, scale="adjr2") ### Resultados baseados no R2 ajustado
plot(all_reg, scale="bic") ### Resultados baseados no BIC
### Nesses gráficos avaliamos os resultados dos critérios para os melhores
### modelos ajustados com cada número de covariáveis.

s1 <- summary(all_reg, matrix.logical=TRUE)
s1 ### A matriz lógica permite identificar as covariáveis selecionadas 
### em cada modelo.

s1$rsq ### Valores de R2 para cada um dos modelos selecionados
s1$adjr2 ### Valores de R2 ajustado para cada um dos modelos selecionados
s1$bic ### Valores de BIC para cada um dos modelos selecionados

which.max(s1$adjr2)
coef(all_reg, id = 12)
### O modelo com doze covariáveis produziu maior valor de R2 ajustado.

which.min(s1$bic) 
coef(all_reg, id = 7)
### O modelo com sete covariáveis produziu menor valor de BIC.

which.max(s1$rsq) 
coef(all_reg, id = 17)
### O modelo com 17 covariáveis produziu menor valor de R2 (obviamente).

### Vamos produzir alguns gráficos usando os resultados desta análise.
n_cov <- 1:17

plot(n_cov, s1$bic, type = 'b', xlab = 'Número de covariáveis', 
     ylab = 'BIC', las = 1, pch = 20)
axis(1,1:17)

plot(n_cov, s1$adjr2, type = 'b', xlab = 'Número de covariáveis', 
     ylab = 'Adjusted R2', las = 1, pch = 20)
axis(1,1:17)

plot(n_cov, s1$rsq, type = 'b', xlab = 'Número de covariáveis', 
     ylab = 'R2', las = 1, pch = 20)
axis(1,1:17)


### Para finalizar, vamos aplicar os algoritmos de seleção do tipo
### stepwise. Primeiramente fixando k = 2, estamos definindo o AIC como critério de
### seleção, temos:

### Método backward
step_back_AIC <- step(aj_full, direction = "backward", data = College, k = 2)
summary(step_back_AIC)

### Método forward. Para o método forward devemos definir o escopo da seleção
### (menor e maior modelo). O menor seria o modelo nulo (apenas com o intercepto),
### enquanto o maior seria o modelo com todas as covariáveis.
aj_lower <- lm(Grad.Rate~1, data = College)

aj_upper <- lm(Grad.Rate~., data = College)
formula(aj_upper)

step_for_AIC <- step(aj_lower, direction = "forward", scope=formula(aj_upper), 
                     data = College, k = 2)
summary(step_for_AIC)

### Finalmente, o algoritmo que considera tanto exclusão quanto inclusão de
### covariáveis a cada passo
step_both_AIC <- step(aj_full, direction = "both", data = College, k = 2)
summary(step_both_AIC)

### Vamos comparar os ajustes.
data.frame(compareCoefs(step_back_AIC, step_for_AIC, step_both_AIC))


### Agora fixando k = log(n) (critério BIC):

### Método backward
step_back_BIC <- step(aj_full, direction = "backward", data = College, k = log(nrow(College)))
summary(step_back_BIC)

### Método forward. Para o método forward devemos definir o escopo da seleção
### (menor e maior modelo). O menor seria o modelo nulo (apenas com o intercepto),
### enquanto o maior seria o modelo com todas as covariáveis.
aj_lower <- lm(Grad.Rate~1, data = College)

aj_upper <- lm(Grad.Rate~., data = College)
formula(aj_upper)

step_for_BIC <- step(aj_lower, direction = "forward", scope=formula(aj_upper), 
                     data = College, k = log(nrow(College)))
summary(step_for_BIC)

### Finalmente, o algoritmo que considera tanto exclusão quanto inclusão de
### covariáveis a cada passo
step_both_BIC <- step(aj_full, direction = "both", data = College, k = log(nrow(College)))
summary(step_both_BIC)

### Vamos comparar os ajustes.
data.frame(compareCoefs(step_back_BIC, step_for_BIC, step_both_BIC))